No colo company wants them as a customer and it would take months to order and install ~200 servers.

Probably because their workloads haven't stabilized at any meaningful threshold. You can only justify the capital expenditure if you know you can amortize it over time. Hard to do for a startup whose existence is being threatened. They are also a political hot potato, and would need the colo provider to agree to take them on and withstand the public pressure, which I doubt most would want.

Their proposed current hardware costs are a tiny capital outlay. Future current hardware costs are a tiny capital outlay.

They will pay far more for bandwidth than they ever will for hardware.

They probably don’t have the personnel who know how to order, provision, and physically manage that kind of hardware. They’re a small startup, and going with AWS means that they probably don’t have any of the necessary skills to do that.

I don't necessarily disagree with the hardware capex vs bandwidth tradeoff, but I don't think either of us are aware of their balance sheet to reasonably claim if this is an insurmountable cost for the business today or not. Paying for several thousand servers, getting them up and running, and tuning them isn't exactly a trivial effort... and then you have to deal with bandwidth.

From the tweet, looks like 540 servers max at this point. And sure, not trivial but you shouldn't be planning to run a large chat system without the technical staff to pull something like that off, whether on AWS or not.