Ok, this is probably a stupid, very bad, no good idea considering I've not heard of people doing this, but can't you retain many of the benefits of numerical IDs but also the secrecy of UUIDs by using an HMAC ?

With HMAC, you can still ask for some sequential IDs

SipHash128(0, KEY) = k_0

SipHash128(1, KEY) = k_1

You get the same number of bits as a UUID.

You can't, however, sort by IDs to get their insertion sequence, however. For that you'd need something like symmetric encryption but this is already a bad idea, no reason to make it worse.