Can we log why a client gets bounced to different backend servers?

There is no way to log why there was new load balancing decision. You can check the logs around that time, and if the pool member was down, you can easily conclude the new load balancing decision was because of that.

You could have an iRule logging a lot of information for every request, but that is not the correct way.

The correct way is to take a tcpdump while you replicate the problem, and analyze that after.

Also, important to mention that a load balancing decision occurs per connection by default, and not per HTTP request.

This solution explains the difference between load balancing per connection or per request.

https://support.f5.com/csp/article/K7964

If you want to play with tcpdump, here is the solution:

https://support.f5.com/csp/article/K411