Round Robin can get weird when you have common factors between the number of subrequests for a single request and the number of servers. I’ve seen everything from certain pages reliably blowing up in prod to weird caching bugs that don’t trigger in preproduction because the same box sees all of the requests.
Developers have a nasty habit of assuming a glitch or someone else changed something and if a reload fixes it they don’t note the pattern.
In load balancing they call this persistence. You are describing a case where you should persist to the server you land on.
In this scenario, unless each request is exactly the same you will end up with uneven load as some sessions persist longer than others.
This can still be stateless with use of a cookie for protocols that support it but is not the idealized "round robin and every request takes same time and resources".
The other alternative is a stateless app that is architected for round robin lb.
Developers have a nasty habit of assuming a glitch or someone else changed something and if a reload fixes it they don’t note the pattern.