Thanks to @StackzOfZtuff for the linked articles. Using those, this is what I've pieced together by now:
The generated parameters can probably be trusted as long as they are regenerated frequently. For a higher security, one would have to check for reused parameters, which could be a sign that something fishy is going on. It's still safer to generate parameters on ones own machine.
As the name suggests, safe primes work well when generating DH groups, as they guarantee a certain size of the resulting group. So deliberately bad primes, resulting in reduced group sizes, are not a problem with the scheme described in the question.
The one possible attack I see is that the generating entity could chose specific groups and precompute as much as possible to solve the discrete logarithm problem using the natural field sieve. This would make it a lot simpler for an attacker to break any DH using that group. Naturally, the precomputation step is computationally expensive, so the attacker would want to reuse any broken group as much as possible. This could be detected by keeping track of previous groups.
Instead of doing checks for recurring groups, just calculating one's own parameters is probably simpler. Using DH parameters generated by others, without any nothing-up-my-sleeves guarantees, requires some trust in the entity providing those new groups, regardless of whether the underlying primes are safe or not.