A failed integration because missing registry secret does not show the error reason

### Bug description

The operator shows the error:
```
camel-k-operator-6c9d9ddf9c-p8pqj camel-k-operator {"level":"info","ts":"2026-08-27T06:59:12Z","logger":"camel-k.controller.integration","msg":"Integration error","request-namespace":"default","request-name":"test","api-version":"camel.apache.org/v1","kind":"Integration","ns":"default","name":"test","reason":"Error","error-message":"Back-off pulling image \"10.109.141.209/camel-k/camel-k-kit-da7tvrmbngqc73e29fdg@sha256:9059e1b787f21baaab273f7ab2fcea42b6a068847efe9ab2bcdece80c3bee03d\": ErrImagePull: Error response from daemon: Get \"http://10.109.141.209/v2/camel-k/camel-k-kit-da7tvrmbngqc73e29fdg/manifests/sha256:9059e1b787f21baaab273f7ab2fcea42b6a068847efe9ab2bcdece80c3bee03d\": no basic auth credentials"}
```
however this is not reported into the Integration status, which would be good for user to know why the Integration is failing.

### Camel K or runtime version

_No response_

Start in the camel-k.controller.integration error path that emits the "Integration error" log, then trace how failures are written to Integration status. Reproduce the missing-registry-secret image-pull failure and verify that its error reason is visible in the Integration status when the fix is done.

Written by the indexing model from the issue text.