Yes after writing my note I did realize that that’s the obvious answer to “isn’t a private version kinda the same?” is: It isn’t the same, for that reason.
But there is a (slightly weak) answer to that: if proofs have to be generated at intervals, e.g. let’s say we are optimistic about performance of this approach and imagine it’s OK to have a 24 hour update cycle, you could require everyone to reproduce proofs based on utxo snaphots on that cycle. Maybe that’s unrealistic, I wouldn’t be at all surprised, for a very large network like Lightning. But there are presumably all kinds of finesses/details here, like what you mention in the remainder of your post.