Your method of generating ids at the server, in the application, in a dedicated request-response, is a very good one! Uniqueness is very important, but clients, like suitors, are going to keep repeating the request until they succeed, or until they get a failure they're willing to accept (unlikely). So you need to get uniqueness from somewhere, and you only have two options. Either the client, with a GUID as Aurélien suggests, or the server, as you suggest. I happen to like the server option. Seed columns in relational DBs are a readily available source of uniqueness with zero risk of collisions. Round 2000, I read an article advocating this solution called something like "Simple Reliable Messaging with HTTP", so this is an established approach to a real problem.
Reading REST stuff, you could be forgiven for thinking a bunch of teenagers had just inherited Elvis's mansion. They're excitedly discussing how to rearrange the furniture, and they're hysterical at the idea they might need to bring something from home. The use of POST is recommended because its there, without ever broaching the problems with non-idempotent requests.
In practice, you will likely want to make sure all unsafe requests to your api are idempotent, with the necessary exception of identity generation requests, which as you point out don't matter. Generating identities is cheap and unused ones are easily discarded. As a nod to REST, remember to get your new identity with a POST, so it's not cached and repeated all over the place.
Regarding the sterile debate about what idempotent means, I say it needs to be everything. Successive requests should generate no additional effects, and should receive the same response as the first processed request. To implement this, you will want to store all server responses so they can be replayed, and your ids will be identifying actions, not just resources. You'll be kicked out of Elvis's mansion, but you'll have a bombproof api.