Improve default logging

[ahg-g]

What would you like to be added:

Currently we pretty much don't log anything since the default log level is 1, which pretty much make it impossible to debug. To improve the default logging, we need two things:

1. Increase the log level to 2 (aka DEFAULT)
2. Audit the log lines to ensure that DEFAULT is used in a sustainable way (for example, a log line indicating probing it being done, but not at the same rate of probing for example)
3. Ensure there is at most two log lines per request, one when receiving the request and one when done processing it (success or error)

Why is this needed:

To improve debuggability

[liu-cong]

+1

BTW this is our logging guideline https://github.com/kubernetes-sigs/gateway-api-inference-extension/blob/main/docs/dev.md

For request logging we have the following.

logger.V(DEFAULT)
* Logging the status code of an HTTP request
* Important decision making such as picking the target model, target pod

As part of this task we should audit existing logging and make sure they align with the guidelines (or update guidelines where it's appropriate).

[zetxqx]

/assign

[nirrozenbaum]

for completeness - pay attention that some information shouldn't be logged, like the prompt itself.