But if Theo were to grant me access, but disallow e.g. putty, that would be well within his rights, no?

No, what software you use is none of his business.

What software I use is none of his business right up until it starts affecting computers he owns. This isn't that complicated. If I decide that I don't want people wearing red shirts to enter my house, then that's stupid, but within my rights and does not meaningfully infringe upon your right to wear whatever color shirt you want.

I don't see how restrictions on what is allowed to connect to privately run servers can magically cause software to become non-free. Nor do I see any way in which those restrictions existing (again on computers neither you nor I own) in any way violates the spirit of free software or in any way denies you and I the freedoms that free software is supposed to respect.

Your house, a personal area, is in no way comparable to a public messenger network.

> What software I use is none of his business right up until it starts affecting computers he owns.

Which client implementing the Signal protocol you use does not affect him, therefore it's none of his business.