Re : "No, I don't want to reconstruct the key", I don't really understand; I didn't use the word 'reconstruct' in that. "I just want these providers to construct sigs safely which apparently FROST can do". But the point that's at issue here is you want them to do that with *your* key. FROST and MuSig both allow multiple parties to collaborate to create signatures safely, with the main difference being that the former allows it for t of n and not only n of n. *Neither* provide for the key to come from some central party. As for distributed key gen, yes they both do it, they have to; but they do it in radically different mathematical ways (because of the t of n requirement for a start). And retrofitting them to allow you to start with a key "x" and then split it up and give it to them probably can work, but my caveat point is, if you care that they can collude to take the whole key, between them, then that's not great.** In FROST's case, it happens to be polynomial evaluations that you share, but it's still designed so that t of those shares (in a t of n scheme) are enough to reconstruct your signing key.

**In any such collaborative signing scheme, by definition, the key can be reconstructed by the authorised threshold t, because they can share their nonce contributions and extract the private key using the nonce. This isn't usually discussed, but here we have an external party that'd prefer not to be attacked by the group, so to speak.