The people who say no thanks to both do not consider those two options as unique in any way. Trump and Biden are inevitable and next election will have an even worse choice. The bourgeois state is declining and in its decline it turns towards fascism. Both parties are extensions of the bourgeois state.
By voting all one doesfurther entrench the system. For a long time the arguments for voting for democrats were:

• You have to just this once it's the most important election of all time
• Harm reduction/"lesser evil"
• This gives us more time to organise for next election.
• The US left can organise better under democrats

As time has passed these arguments have shown themselves to be based on some faulty assumptions. Or conditions have changed to such an extent that the argument no longer stands.

• "Just this once"

1. It has been "the most important election" since, at least, Reagan - With the sole exception of Clinton v. Dole. And in retrospect the hysteria seems much less worrisome. Obama ran against McCain and Romney, respectively. Romney was made out to be a religious fanatic with "binders full of women" who hated poor people "47%". McCain was a senile warmonger with an insane VP. Both of these were quickly rehabilitated by the establishment, in death McCain was grieved by his political opponents. Dubya has been rehabilitated into the image of a quirky old man. Most democrats claim to look up to Reagan and will quote him. They almost all pursue "reaganomics", which Reagans VP had referred to as "Voodoo economics" before adopting it himself. Good ol' trickle down. So it is hard to believe it will be "just this once" and it is hard to believe that there is anything especially egregious about this specific republican, when this is what is always said.
2. On the other hand we can accept that every single election has been the most important election ever and we need to vote democrat or lose democracy forever. If this is really the case, then one has to wonder if a system where we don't actually have a vote - It's democrat or death - is worth perpetuating. One also has to wonder why the democrats, when they are in power, do not dismantle the republican party - If the republicans truly are fascists opposed to democracy. One has to wonder why democrats constantly seek to work with the republicans and speak about "bipartisanship". The conclusions available all lead to the same place: The democrats are either complicit or completely inefficient. Thus a vote for a democrat is either inconsequential or actively harmful.

• Vote for harm reduction/lesser evil

This argument functions in the short term. However every election there's less and less "lesser" evil. Wether they win or lose democrats make the same conclusion: They must go further to the right. The democrats are also just throwing marginalised groups under the bus (This is ignoring that democrats have been doing all the "evil" stuff on a state-level for much longer than their presidential platform reflects). Every administration since, at least, Clinton has deported more immigrants than the previous administration. Biden surpassed Trump 1 ^[Some will protest that Biden did not "deport" more immigrants, he technically "removed" them instead. Technicalities like these are moot in my eyes, they function only to illustrate how democrats run the same child-crusher, they're just more "polite" about it. The child is crushed wether it is crushed or "pulverized".] They both suck on climate^[Some will protest that Biden passed the Green New Deal or that he signed the Paris Agreement. The Paris Agreement is a pointless treaty and he did nothing to approach its' goals. The "Green New Deal", while allocating some resources to renewable energy, was in practicality nothing. The actual development of the US under Biden was one where greenhouse gas emissions increased and one where he gave out so many fracking contracts that the emissions from the oil in those alone will be enough to push us up to 3.5C warming. It does not matter that you are kissing my booboo, if you're still stabbing me. The climate is not something that can be half-measured, it is an apocalypse and it is either-or. It needs IMMEDIATE and DRASTIC action, something neither party is willing to take. Therefore the faster neither party is in power, the better.], they both commit genocide, they are both expanding a police state, they're both imperialists, they're both giving more money to the military.
So while one side might give slightly less, the long term result of this "lesser evil" is one where more harm is produced, than the long term result of one where the system is replaced^[Here is an admitted assumption. I assume the system is easier to replace if it is less entrenched. This seems obvious to me, but others have wanted to discuss this a bunch. It's a discussion in itself.].
What do you think will happen when not even genocide is too far?

• Buying time to create an alternative

1. By voting for democrats, it is harder for an alternative to seem viable, since the democrats appear to have broad support.
2. This is an argument built on thinking long term. However when someone suggests an alternative to voting for democrats, they are challenged on the immediate viability of such an action. "There is no alternative right now" or "Will not voting democrats make things better right now?" or "What will voting third party do to oppose [villain of the election] right now?"
   If we accept that the argument for voting for democrats is, in part, built on longer term planning, then we must likewise accept the longer timescope for arguments against voting democrat.
3. Democrats also crack down on left-wing organisers. Democrats also crack down on democratic movements. There are many examples, but most preciently might be Joe Bidens crackdown on anti-israel protesters or his violent crackdown on the remains of "the summer of ´20" once he came into power. He continued Trumps methods of having people disappeared in unmarked vans. It was under democratic governorship and presidency that ferguson riot leaders where assassinated and disappeared. Democrats destroy organisations that are even beneficial to themselves
4. Democrats functions a sheepdogs for the left. They are the graveyard of social movements.

• The US left can organise better under democrats

I don't know if conditions are "better" for organising. That's hard to say in my eyes. We have seen far more popular unrest under Trump. People are far more displeased with the regular actions of the US empire under Trump. People have become far more radicalised under Trump. The popular movements are not funnelled into or through the DNC, but instead develop organically. There are other challenges under Trump than under democrats, but "better" or "worse"? Hard to say. Internationally I would consider Trump better, because he is more incompetent.

The same commentators who through introspection can say that socialist movements across the world need to work with the USA isnt a betrayal of the revolution seemingly cannot apply the same caveated analysis to the shinier of two turds in a two party system sounds like more American exceptionalism.

This feels like a bit of a different discussion. The actions, motivations and needs of revolutionary movements engaged in military struggle around the world are different from that of the average US voter (as well as the average US leftist).
You are also conflating "voting for a democrat or a republican" with "agreeing to receive support from the US." It's kind of vague, so I don't really know what you're talking about here and tbh it is not my area of expertise. I'm going to send a link to this comment to someone else, who might be better at answering, but I'd really appreciate more specificity. Can you say what socialists movement across the world are you thinking of and what socialist commentators are you talking about?
From what I can think of: I know many people have critiqued the various Kurdish militias for cooperating with the US, the PRCs cooperation with the US during the sino-soviet split is seen by most as a massive black mark on that countrys history.
In general though I'd say that people are more forgiving of foreign movements accepting US aid because the aid helps them further towards their goals, whereas someone voting for the democrats would, at best, do nothing. The way I view it, it would make things worse for the domestic leftist to vote democrat. I can't speak on local politics.

If you're not voting for your “communist party of”, is this just virtue signalling?

I am voting third party. Here is a comment I wrote about leftist not voting or voting third party that whole thread has many illustrativt responses.
Here is another comment from the post, detailing my position
Here is my comment from another post
I would also recommend reading cowbees response in that comment thread.