Integration tests does not wait server properly with `stay_alive=True`

Even after I disable async logging it is still ends with

```
2026.01.09 10:12:42.936111 [ 231 ] {} SystemLog (system.query_log): Terminating
2026.01.09 10:12:42.936210 [ 230 ] {} SystemLog (system.part_log): Terminating
2026.01.09 10:12:42.936279 [ 232 ] {} SystemLog (system.background_schedule_pool_log): Terminating
2026.01.09 10:12:42.936326 [ 233 ] {} SystemLog (system.trace_log): Terminating
2026.01.09 10:12:42.936365 [ 234 ] {} SystemLog (system.crash_log): Terminating
2026.01.09 10:12:42.936453 [ 235 ] {} SystemLog (system.text_log): Terminating
2026.01.09 10:12:43.067652 [ 236 ] {} SystemLog (system.error_log): Terminating
2026.01.09 10:12:43.068062 [ 237 ] {} SystemLog (system.s3queue_log): Terminating
2026.01.09 10:12:43.068140 [ 238 ] {} SystemLog (system.query_views_log): Terminating
2026.01.09 10:12:43.068216 [ 239 ] {} SystemLog (system.asynchronous_insert_log): Terminating
2026.01.09 10:12:43.068311 [ 240 ] {} SystemLog (system.backup_log): Terminating
2026.01.09 10:12:43.068388 [ 241 ] {} SystemLog (system.blob_storage_log): Terminating
2026.01.09 10:12:43.068457 [ 242 ] {} SystemLog (system.query_metric_log): Terminating
2026.01.09 10:12:43.068530 [ 243 ] {} SystemLog (system.zookeeper_connection_log): Terminating
```

Interestingly though that I cannot reproduce it w/o integration tests, but commenting `shutdown_system_logs` helps