Here is the response I've got from the support:
The calls to _lastCheckTime.txt are made by the storage health checker from App Service. This is necessary because if the share is unhealthy for long enough, we need to try to auto-recover.
The health checks are being counted as 3 operations (create, queryinfo, close).
The health check I/O operations performed by the platform are translated into three distinct operations on the storage account. The platform mounts Azure Files shares using the CIFS driver which uses the SMB protocol to interact with your file share, and performs I/O checks using standard .NET file system libraries, so the engineers do not have direct control over the translation of these operations to the corresponding storage API requests.
These health checks are necessary to ensure that running applications maintain consistent access to the mounted storage, and to allow the platform to attempt automatic recovery if the mount becomes inaccessible.