[source-productboard] v1 API retired by Productboard, migrate to v2

### Connector Name

source-productboard

### Connector Version

0.0.63

### What step the error happened?

Configuring a new connector

### Relevant information

Productboard has **retired the v1 REST API**. `source-productboard` calls v1
endpoints with the `X-Version: "1"` header, so **every sync on this connector
now fails**

[productboard_error_logs.txt](https://github.com/user-attachments/files/30351593/productboard_error_logs.txt)

this affects all users of the connector, not a single workspace.

Migration guide: https://developer.productboard.com/reference/migration-guide

### Relevant log output

```shell
2026-07-24 12:01:06 source ERROR Max retry limit reached after 10.5s. Request: , Response:
2026-07-24 12:01:06 source ERROR Retries exhausted with backoff exception.
Traceback (most recent call last):
File "/usr/local/lib/python3.13/site-packages/airbyte_cdk/sources/streams/http/http_client.py", line 292, in _send_with_retry
response = backoff_handler(rate_limit_backoff_handler(user_backoff_handler))(
request,
...<2 lines>...
exit_on_rate_limit=exit_on_rate_limit,
) # type: ignore # mypy can't infer that backoff_handler wraps _send
File "/usr/local/lib/python3.13/site-packages/backoff/_sync.py", line 105, in retry
ret = target(*args, **kwargs)
File "/usr/local/lib/python3.13/site-packages/backoff/_sync.py", line 105, in retry
ret = target(*args, **kwargs)
File "/usr/local/lib/python3.13/site-packages/backoff/_sync.py", line 105, in retry
ret = target(*args, **kwargs)
File "/usr/local/lib/python3.13/site-packages/airbyte_cdk/sources/streams/http/http_client.py", line 386, in _send
self._handle_error_resolution(
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~^
response=response,
^^^^^^^^^^^^^^^^^^
...<3 lines>...
exit_on_rate_limit=exit_on_rate_limit,
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
)
^
File "/usr/local/lib/python3.13/site-packages/airbyte_cdk/sources/streams/http/http_client.py", line 541, in _handle_error_resolution
raise UserDefinedBackoffException(
...<5 lines>...
)
airbyte_cdk.sources.streams.http.exceptions.UserDefinedBackoffException: Response was not ok: `410 Client Error: Gone for url: https://api.productboard.com/products`. Response content is: {"message":"The Productboard API v1 has been retired for this workspace. Migrate to API v2 — see https://developer.productboard.com/reference/migration-guide"}
```

### Contribute

- [x] Yes, I want to contribute

---
**Internal Tracking:** airbytehq/oncall#13152