Let's start with a 2010 quote by Walter Mead:
A group like AIPAC enjoys power and recognition not because it controls or even represents the votes of Jews. AIPAC’s power rests on gentile ideas and support; if a politician gets loudly and publicly labeled anti-Israel by AIPAC and its allies that politician will get hammered in the next election because so many American gentiles want their politicians to support the Jewish state. AIPAC works like the NRA; it is the publicly accepted voice on an issue about which the public has strong views.
Now, Mead has staked out a position roughly opposite that of Mearsheimer (author of The Israel Lobby and U.S. Foreign Policy). To quote that (2007) book's Goodreads entry:
... it (Israel Lobby) provoked both howls of outrage and cheers of gratitude for challenging what had been a taboo issue in the impact of the Israel lobby on U.S. foreign policy ... remarkable level of material and diplomatic support that the United States provides to Israel and argues that this support cannot be fully explained on either strategic or moral grounds. This exceptional relationship is due largely to the political influence of a loose coalition of individuals and organizations ... *
wikipedia's coverage of reactions to the book stated (remember that Wikipedia coverage of politically sensitive subject can be less than wholly neutral):
Mark Mazower, a professor of history at Columbia University, wrote that it is not possible to openly debate the topic of the article: "What is striking is less the substance of their argument than the outraged reaction: to all intents and purposes, discussing the US-Israel special relationship still remains taboo in the U.S. media mainstream. [...] Whatever one thinks of the merits of the piece itself, it would seem all but impossible to have a sensible public discussion in the U.S. today about the country's relationship with Israel."
or
Stephen Zunes, professor of politics at the University of San Francisco, gives a detailed point by point critique of the paper.[45] Zunes also writes that "The authors have also been unfairly criticized for supposedly distorting the history of the Israeli-Palestinian conflict, though their overview is generally quite accurate," and agreed with Joseph Massad's interpretation of Mearsheimer's and Walt's argument: "[T]here is something quite convenient and discomfortingly familiar about the tendency to blame an allegedly powerful and wealthy group of Jews for the overall direction of an increasingly controversial U.S. policy."[45]
Mead's dispute with Mearsheimer
Mead also notes that contrary to Walt and Mearsheimer's claim that pro-Israel groups exert influence through campaign finance, pro-Israel groups contributed less than one percent of PAC contributions in the 2006 election cycle. Mead agreed that pro-Israel political advocacy is a topic worthy of study but argued that the US policy on Israel grows out of more diverse and complicated historical reasons than described in The Israel Lobby.
With that out of the way...
The USA is a nation of diasporas, "moral certitudes" (the US is always right) and, yes, lobbies.
Yes, a diaspora feeling very strongly about a single issue can influence policy. One such has indeed largely steered foreign policy on a narrow subject, presumably decided a POTUS election and flipped what used to be contested state into solidly a one-party state.
AIPAC? No, the 2000 POTUS election, Florida and Cuban policy, via the Cuban expat community. Do we hear much muttering about Cubans ruling the US? We do not.
AIPAC, and the Cuban expats operate standard, legal, lobbying entities. With the electoral transparency they are required to use, no more, no less. They do not control US foreign policy, but, as per Mead, they do manage to steer policy in a way that both suits them and already resonates with the US public (note that Mead wrote this in 2010, the popularity of AIPAC is not written in stone). That is what lobbying organizations do.
Generally speaking, Jewish people (7.6M / 2.4% US population) are well-regarded in the US and, as Israel is also, for reasons already asked about on this site, held in fairly high esteem, their support for Israel is not contentious. If the Cuban expats were instead supporting Castro's legacy, they presumably would not be well regarded. China's expats, if they were to deploy a charm offensive to support Xi, would be villified. If those same expats attack the CCP, they will be applauded. The starting context matters: does the US public like the policy advocated?
Likewise, Biden reminded the UK PM a while back that screwing up the Good Friday accords wrt Ireland's land borders would not be taken kindly in US trade deals. People are expected to support their roots, within reason, as long as that support "flows" with general US sentiment.
What about Palestine?
Sad to say, when Gaza makes the news it usually starts with the wrong reasons. Americans react badly to terrorism. They don't pay that much attention to the population's day to day hardships, hold the region in low esteem (no democracies like ours there!) and generally avoid looking too closely at what Israel is up to.
If the Palestinian people had better representatives, they would likely get better press. As it is, the spokespersons are Hamas and the PLO's not-recently-elected Abbas. Absent mass civilian casualties their sympathy meter just does not stack up to Israel's.
Who else supports Israel?
Some of the strongest supporters of Israel are US Evangelicals. They most certainly are not held under thrall by AIPAC.
To reiterate: the nature of AIPAC's influence is not a problem by itself: it remains well within the laws and customs of lobby groups, even if it is certainly quite influential. The goals of AIPAC, which seems largely to be to insulate the Israeli government from having to compromise is a problem, to some. If J Street, a much smaller, pro-peace Jewish American lobby group was ascendant, there would be no complaints (from those wishing for a 2 state solution).
Does it matter?
Traditionally, Presidents have steered clear of taking on Israel, even on fairly clearcut issues like the illegal settlements, until they are near the end of their second term and do not have to face reelection. Bush senior, Clinton, IIRC.
Or, Guardian:
The groups are a powerful force in US politics that draw comparisons to the National Rifle Association (NRA) at the peak of its power, and spent more on the 2022 Congress than other special interests, such as the oil and gas industry.
The former president Barack Obama, in his 2020 memoir, detailed the threat Aipac presents to Israel’s critics, who risked “being tagged as ‘anti-Israel’ (and possibly anti-Semitic) and confronted with a well-funded opponent in the next election”, he wrote.
As to the fact that AIPAC's influence is discussed and hardly a secret, I defer to another answer.
The big honking elephant in the room
So far I've compared AIPAC to the Cuban expat lobby, with comparisons to Chinese and Irish-root expats as well.
But that is not the whole story. The Protocols of the Elders of Zion is a long-standing conspiracy theory, started in early 1900s Russia, exploited in 1930-1940s Nazi Germany and part-resurrected at a UN convention in 2003. The Protocols claims that Jews "control stuff" and is a recurring staple of anti-semitism at its most virulent. In general "Jews controlling too much stuff" is a bog standard anti-semitic talking point.
AIPAC gets little challenge because
it operates legally and in similar fashion to other lobby groups
its aims, for better or worse, are in line with US public sentiment. And those of its opponents, less so.
criticism of it can, rightly or wrongly, depending on the nature of said criticism, be associated with some of the most vile anti-semitic crap humanity has been able to drum up.
It should be possible to question US foreign policy wrt Israel, like other branches of its foreign policy get questioned, but the anti-semitism stink is a strong deterrent. And so is the tendency of pro-Palestinian supporters to pooh-pooh Hamas' atrocities.
* I rather liked Mearsheimer's book, which is definitely a work of realpolitik. I think it's a bit easy to accuse him of anti-semitism in it, he mostly seems to object to AIPAC's aims not aligning all that much with US interests, rather than its methods, and would not mind a stronger voice from say, J Street. That said, Mearsheimer's position on Ukraine, that NATO left Putin no choice, has definitely soured me on his analysis and I wonder what intellectual shortcuts he took in writing the Israel Lobby.
p.s. while the question can certainly be considered contentious, not least for its Protocols-y whiff, the massive downvoting it attracted does somewhat support the notion that it bears asking. Is there enough realpolitik assessment of what supporting Israel come-what-may gains the US? Or indeed Israel itself, if it has to continue its present policies indefinitely? Then again, how much does the US-Cuban foreign policy really get questioned?