in order to study cognition we have to take the body into consideration.

I agree. I don't think embodiment is irrelevant; my own field (human-computer interaction) takes embodiment quite seriously — it's an absolutely integral factor in natural user interface design, for example.

I just don't think embodiment is in any way magic, the way that the embodied cognition people seem to think and imply. If you can simulate a human and their environment on any level you like, then embodiment stops being an issue. It seems like we don't actually disagree on this.

I don't like to make assumptions about how subjective experience works, and for all I know, it arises from some subtrates of cognition but not others.

This is certainly not impossible, but it's not clear to me why you couldn't then simulate the substrate on a sufficiently low level as to capture whatever aspect of the substrate is responsible for enabling cognition. After all, we could in principle simulate the entire universe down to quantum configuration distributions, right?

If you wanted to make a weaker claim based on computational tractability, then that would of course be another thing.

Since I think of my subjective experience as an essential part of my self, this seems important.

I concur with this. To the extent that I have any kind of a handle of what subjective experience even is, it does seem quite important.

P.S.

Hmm, I can see arguments for and against calling computationalism a form of dualism. I don't think it matters much, so I'll accept your claim that it's not.

Yeah, this is probably a question of preferred terminology and I am not inclined to argue about it too much; I just wanted to clarify my actual views.