Keys ain't balls:
Crypto keys are unique and never reused, no "putting back" like your urn nonsense.
Balls are not the keys, they are the hashes. The keys are irrelevant here. They might as well be random data that gets hashed. This doesn't change in any way any probabilities whatsoever.
Real-world:
Prefix search actually saves work by bailing early when collisions happen. Your urn keeps uselessly drawing balls forever.
No, no. The urn does the same bailing just as the prefix search: extractions are skipped once a blue ball comes out. This step is very important, sorry that I forgot to mention it. So, please, once a blue ball comes out, please skip the next few extractions. Then please proceed with the next extraction.
Physical constraints matter:
The prefix search exploit actual hardware behavior (sequential scanning + early termination) to boost efficiency. Your model just jerks off in theory land.
My bad, I totally forgot how much that matters when doing the probability calculations. It definitely impacts the next ball to come out. IMHO I was thinking it's exactly the other way around: not skipping keys helps because you simply compute the next key, instead of a hard reset. My bad again.
Have fun!