I found via politics.SE that MRC-owned site Newsbusters claims that:

In a discussion of modern electoral fraud, MSNBC’s Andrea Mitchell on Monday offered a journalistic admission: She conceded that the 1960 presidential election was “obviously” “stolen” from Richard Nixon. [...]

She continued, providing this example: “Richard Nixon not challenging... that Kennedy had stolen the election in 1960, which obviously had been stolen in 1960.” It’s rare for liberal journalists to admit the electoral fraud of the dead voting for Kennedy in Illinois. (Voter fraud also occurred in Lyndon Johnson’s Texas.) JFK’s popular vote margin was just over 100,000 votes out of 70 million cast. [...]

Now that 56 years have passed, more journalists have openly talked about the fraud in the 1960 campaign. In the 2016 CNN documentary Race for the White House, ex-Newsweek editor Evan Thomas noted, "As time goes on, it leaks out that, in one black district, there were more votes cast than there were people living in the district. It was corrupt." Narrator Kevin Spacey explained, "As the dead of Illinois cast their votes for Kennedy, there are more allegations of fraud in Texas."

And a bit more googling found an opinion piece in Newsweek:

Pulitzer-winning journalist Seymour Hersh reported hearing tapes of FBI wiretaps about potential election fraud. Hersh—whose books indicate he is a fan of neither Kennedy nor Nixon—believed Nixon was the rightful winner.

So is there some kind of historical consensus now that JFK's election win was due to fraud?

No, there is not consensus that the 1960 election outcome was affected by fraud

You ask (emphasis mine):

So is there some kind of historical consensus now that JFK's election win was due to fraud?

If there is a controversy, there is by definition not consensus.

Wikipedia has an extensive section about the controversies regarding the 1960 US Presidential election. There are plenty of references there to show that there are several controversies regarding the election, and with that we have enough to settle the question: no, there is not consensus.

Regarding more votes than registered voters:

This analysis, though, ignores the fact that the registered voter figures measured only individuals who paid the poll tax and that certain groups were exempt from having to pay that tax.

Reference: https://www.chron.com/opinion/outlook/article/Vote-ID-law-is-a-waste-of-money-3481335.php

In 1960, the Nixon presidential campaign charged that Fannin County, Texas, allowed more people to vote in that year's election than had paid poll taxes, the unconstitutional $1 to $2 ballot box admission fee once mandated by the state. Indeed, 6,138 ballots were cast in Fannin County when only 4,895 people had paid the poll tax. No charges were ever brought because it appears that much of the "fraudulent" voting may have been committed by people exempt from paying the poll tax: veterans and senior citizens and some other isolated groups.

So there was no fraud?

We cannot say for sure, herein lies the controversies. But since every modern US election have over a hundred million eligible voters, it would be extremely unlikely that there were not at least some instances of it. The extent of supposed fraud though is not known with certainty.

But was the election outcome affected by fraud?

We cannot say that either. Note though that Nixon would have had to win over both Texas and Illinois to secure the presidency. So whoever wants to claim that the election was "stolen" by fraud has to prove there was sufficient fraud in both those states to win them over to Kennedy from Nixon.

Looking at the results, JFK's majority in Illinois was 8,858, or 0.2% of the vote. There could be doubt about this, certainly. However as MichaelK notes, Nixon would have had to win both Illinois and Texas to change the final result.

In Texas, Kennedy's majority was 46,257, or 2% of the vote. Regardless of whether some fraud took place (and realistically we cannot ever fully eliminate that in any vote), this would have needed such substantial vote-rigging as to be very obvious. There doesn't appear to be strong evidence for fraud on this scale.

Also regarding a minor point in your quote, due to the Electoral College system it's immaterial which side wins the popular vote, as exemplified by the elections of Donald Trump (2.1% majority for Hilary Clinton) or George W Bush in 2000 (0.5% majority for Al Gore).

There are some factors that other answers have not covered.

The main claim centers around Illinois. Even though flipping Illinois would not have changed the overall result, there are some things to consider.

First of all, let's concede that Chicago politics is a filthy/dirty as it gets (Report:Chicago most corrupt city in the U.S), and was/is largely controlled by the Democratic political machine, and that Mayor Daly controlled things as much as any politician in recent memory. Does that mean that it's a given that Illinois was stolen? No.

His father, the late Mayor Richard J. Daley, built the once-mighty machine that doled out jobs and favors in exchange for support for Democrats on Election Day. He was never charged with criminal wrongdoing, but several of his high-ranking aides were sent to prison for political patronage.

NBC News, Crime and Courts:Illinois has a long legacy of public corruption

Kennedy won the Chicago vote by a huge margin, but, demographically, that margin was not unprecedented, and Kennedy had certain demographic appeal that was unprecedented -

Kennedy’s margin in Cook County is sometimes overstated as 450,000 votes (or more precisely 456,312), but that figure is limited to the city of Chicago. Cook County also includes a large suburban area, which Nixon won. Across all of Cook County, Kennedy’s margin was 318,736 votes.

For comparison, Dwight Eisenhower had won Cook County by a similar margin in 1956 (315,402 votes). In 1964, Lyndon Johnson would win Cook County by more than twice as many votes (641,463).

Of course, 1956 and 1964 were landslides. How did Kennedy rack up such a margin in a squeaker like 1960? The key is Cook County’s demographics, which were very favorable to Kennedy. Thirty-nine percent of Cook County residents were Catholic, and 20 percent were black. Black and Catholic voters heavily favored Kennedy, who was Catholic and had recently helped to secure Martin Luther King Jr.’s release from a Georgia prison.

There were two Cook County recounts, because the overall results were so close. One was immediate, so the results could be certified, and another was after Kennedy was sworn in. Both narrowed the margin, but neither flipped it. Now, that doesn't mean that there were no shennanigans, but one of the arguments pointing to fraud is the claimed lack of recount.

One way to bring fraud to light is with a recount, and there is a persistent myth that there was no recount in Cook County. In fact, there were two recounts — one in November 1960, before the state vote was certified, and one after Kennedy’s inauguration in 1961.

According to Kallina’s history, both recounts whittled Kennedy’s margin in Cook County, but not by enough to erase his lead statewide.

Dissatisfied Republicans pointed out that some forms of fraud — such as buying or coercing votes — were not the kind that could be corrected by simply recounting the votes. Democrats countered that the downstate counties, which hadn’t been recounted, might have had fraud favoring Nixon, but Democrats did not offer evidence to support their accusation.

WaPost: Here's a voter fraud myth - Richard Daly stole Illinois for Kennedy

Why didn't Nixon demand a recount to rectify the fraud? One item that is accepted by historians, along with the probable pro-Kennedy fraud in Chicago, is that there was also rampant pro-Nixon fraud in downstate Illinois. While a recount might bring the Daly abuses to light, it would also expose similar fraud going on for Nixon, and ultimately would probably not be worth that examination for a result that would not change the outcome of the election.

It is true that Vice President Nixon did not demand a recount in Illinois, despite allegations that Democratic voter fraud in Cook County was rampant, possibly even greater than Republican fraud downstate. Mr. Nixon patriotically stated (after some delay) that his decision recognized the need for national unity and stability during the Cold War. But, to put it in perspective, the following should be noted:

(1) Mr. Nixon decided to refrain from challenging the result only after, as stated in his memoirs, he had "looked into the legal aspects of the situation" and found no basis on which he could obtain a timely victory.

(2) A successful lawsuit switching Illinois from Kennedy to Nixon would not have ended Kennedy's electoral vote majority, even if Alabama's six "faithful electors" had joined their segregationist colleagues in voting for Harry Byrd. Kennedy's Texas margin was too large to reverse, even had a recount been available.

(3) Having trailed in the popular vote on Election Day (depending on how Alabama's vote was measured), Mr. Nixon was ill-positioned to claim, even if some ballots should have been ruled out, that the will of the American people had been thwarted.

WaPost: Not quite 1960