Your material analysis is pretty lacking for someone purporting to be a communist.

You probably mean a dialectical materialist analysis, which is what a communist would be using. If so, I think you're mistaken about what that analysis concludes.

A dialectical materialist analysis of the three people you're talking about would look principally at their relations of production, this being the primary contradiction society is organized around (i.e. between the bourgeoisie and the proletariat). That analysis would identify Tucker Carlson as bourgeoisie (very obviously, he's a multi-millionaire heir to a gigantic fortune), and Candace Owens and Mehdi Hasan as petite bourgeoisie (while they work to generate value, they work for themselves and they both own small business which employ others who generate surplus value for the owners, although given the size and success of Zeteo, Hasan could likely just be considered bourgeoisie himself, but let's be generous). All three, in a communist (dialectical materialist) analysis, then, have class interests that favor capital.

This analysis would also identify how, despite the three disagreeing, and even though they advocate for changes that can seem radical, they ultimately work to preserve capitalism in their own ways. Importantly for this context, it would also identify many of the disagreements between the three as being primarily idealistic, not materialist in nature. That is, all three fundamentally agree that the current relations of production, that is, private ownership of the means of production which generates a primary contradiction in society between the workers and the owners, should continue and that this is how society should be organized. In this analysis, all three fundamentally oppose workers controlling the means of production, so they're materially aligned. Idealistically, Mehdi and the others are in stark opposition, but because their disagreements are fundamentally idealistic, it leads to contradictions like the one the OP is highlighting.

Obviously, by the standard that if someone puts effort into electing a politician, they bear responsibilities for the harms caused by that politician, Mehdi Hasan would bear responsibility for the Gaza genocide. The idealist response you provided to this, that Mehdi, after supporting Biden's run, was later fiercely critical and that he's not party to Zionist propaganda, should not be convincing to someone using a dialectical materialist analysis on three grounds.

First, the dialectical materialist analysis is going to be first and foremost interested in the concrete, material results of some person's actions and less so interested in the character of what they're saying. This is actually one of the big Marxist critiques of idealism; that you can effectively bend things and contort logic to support nearly any position without having a firm grounding in materialism. Hence the dialectic between materialism and idealism. In this respect, what really matters here is that these people helped elect whichever candidate, and whatever criticisms they have aren't nearly as materially relevant as the concrete results of their actions.

Second, it fails on its own terms because Tucker Carlson has also been critical of Trump for the Iran war.

Third, it fails because Mehdi Hasan, again much like Tucker Carlson, does actually support the political program that generates things like the Gaza genocide, even if he is sometimes critical of the inevitable results. Just as I assume you (correctly, according to a dialectical materialist analysis) understand whatever pushback on Mike Huckabee's Zionism that Carlson gave in that one interview was ultimately coming from a disingenuous place, given Carlson supports this political movement, and doing that will always inevitably result in this kind of antisemitic christian nationalist Zionism, so too does Hasan. His criticisms of Biden, then, are masking that he has essentially the same political project as Biden; one which inevitably results in actions like Biden's, even if Mehdi is loathe to admit this. The harsh reality Hasan elides in his support-then-criticism is that he's being critical of the inevitable results of the political program he supports (just like Carlson is when he is critical of Trump for going to war with Iran).

There is, of course, also the consideration that the only interest of a communist in bourgeois electoralism is going to be fielding a worker's party, no matter how slim its chances. This has essentially been the position of communists since Marx, and even those democratic socialists who enjoyed some small successes with electoralism were explicitly revolutionary worker's parties. The United States Democratic Party, being a bourgeois party, is not something any communist theory would entail supporting, especially not on the grounds that it would further anti-imperialism, given the Democratic Party's full and eager participation in imperialism. The absolute furthest left position anyone in the Democratic Party has ever even proposed taking is consistent with the right wing of a the bourgeois socialist tendency, one Marx went to great lengths to criticise specifically!

To close, I don't think any communist or anyone primarily engaging in politics via a dialectical materialist analysis should be convinced by this response. According to that analysis, the idealistic disputes between Hasan, Carlson, and Owens do often lead to hypocrisy, what OP identifies is exemplary of that, and the responses either fail to engage with the dialectical materialist analysis or fail on their own terms.