I have created ticket with MS support and having answer:
I have reached our engineering team and you are correct - this scenario is currently not yet supported. Very few customers use ADLS gen1, so there has been very little demand for feature this so far. ADX does not support refresh tokens, so we cannot renew the token OBO of the service principal. I have sent your feedback on the need of this feature to the engineering team – and for long term (unfortunately no ETA right now), we plan to support using managed identities in continuous export, and then this will be supported.
In the meanwhile, our suggestion is to move to ADLS gen2, where you can configure the export with an account key (or a very long living SAS key).