Is Postgres insert performance the same in the replica
It could improve performance on your INSERT operations (and thus improve performance where those operations occur in the application) depending on how your read replica is setup and if your INSERT contention is partly due to locking caused by heavy reads against your primary database.
I'm not an expert on PostgreSQL replicas in practice, but the theory is applicable, that if your read replica is asynchronously synchronized from your primary, then any SELECT queries executed on your read replica shouldn't cause locking against your primary database (like they normally would if they were executed on the primary). It sounds like it is synchronized asynchronously if it's streaming the WAL which is being asynchronously written to. If that's the case, and if your database was experiencing heavy read lock contention, it's possible you'll see INSERT performance gains on your primary database by moving your SELECT queries to the read replica.
As far as the performance of the SELECT queries goes, I believe you're likely correct, that you won't necessarily see any performance gains since they'll still be bound by the same CPU, I/O, and write lock contention as the read replica is updated.
If it is easy enough to test, I'd say it's worth trying, if you got nothing to lose.