What do you mean by infringement already? So you mean it automatically infringes copyright for all its output just because it might create something similar to a copyrighted work? Or do you mean that if it does create a copyrighted work that work in infringing on a copyright? Your wording is vague here.

can be shown to be capable of reproducing something close enough to said material

I don't think it is a good benchmark for forbidding AI generation of content. If you create a random image generate that has no inputs and is truly random then it is capable of generating something similar to copyrighted work - by pure chance. Even if that chance is very low you could generate enough images and show it can create something similar to copyrighted works.

What happens if you create one that is trained only on public domain images or works properly licensed? Its output is still partially random and could still generate an image similar to some other copyrighted work outside of its training set by pure chance.

I would argue that both of these should be allowed. They are not doing anything obviously wrong even if they could be used to generate copyrighted works. Just like you could use photoshop - or a paint brush to create copyrighted work.

But then, what if you take some other AI that is trained on all sorts of data, copyrighted or not. But then the output of that is fed through a checker that compares it to the training set (and maybe more copyrighted content) and rejects/regenerates work until it is known to not infringe on copyrighted work. Making the chances of it ever producing a copyrighted work far less then the above programs? Should that be allowed? It is using copyrighted work much like an artist would and you could argue that any copyrighted work it does produce was by pure accident as there are intentional steps to mitigate that.

If you use a paid service like Midjourney to generate copyrighted content, the company is essentially selling you access to copyrighted content they lack the rights to.

As far as I understand the laws involved, yeah I would expect that to infringe on some copyright holders work and midjourney would likely be coppable for damages. Just like hiring a artist to create some work and they decide to copy some copyrighted work would also make that artist coppable for damages.

And you also have to consider another side of things - if you can effectively stop AI from training on most works you will effectively stunt its usefulness. Which could lead all efforts in regulated nations to become useless which can result in it just moving to places that are much more open with the technology and where authors of the copyrighted work will have far less control over things. IMO AI generated content is out of the bag now and we will not get it back in. So the best we can do is ensure the right people get compensated for their works. Push to hard in the wrong direction (either way) and there is a real chance they never will.

I don't really have the solutions to many of these problems - but I do think it is worth talking about and don't think that outright bans (or actions leading to an effective ban) on this tech is the correct way to go.