Hello Lukáš,
It looks like you already have the right configuration however if it's not working you can validate with "View my access" option on the storage resource and verify that you have the storage blob data role required. https://learn.microsoft.com/en-us/azure/role-based-access-control/check-access.
Additionally you can check the Storage Blob logs to identify any other issue on your access.(https://learn.microsoft.com/en-us/azure/storage/blobs/monitor-blob-storage?tabs=azure-portal#kusto-queries)
If the above response helps answer your question, remember to "Accept Answer" so that others in the community facing similar issues can easily find the solution. Your contribution is highly appreciated.
Cheers,
Luis