Try commenting the old Mac to the new Mac via . If you can the old Mac will look like an external drive to the new Mac and you can copy whatever files you want to the 4 TB drive on the new Mac.

Then you can not Migrate.

What you want to do is erase and format the SSD to GUID Partition Map/APFS using Dusk Utility.

Erase and reformat a storage device in Disk Utility on Mac - Apple Support

Then manually move your Photos, Music, Movie Libraries to the SSD in drive.

Move your Photos library to save space on your Mac - Apple Support

Change where your music files are stored on Mac - Apple Support

Move your iMovie for Mac library - Apple Support

After that, you can copy documents and other file folders to the Mac mini or SSD.

As for Time Machine backups, those should always be done on a separate external drive and not the same one that your Photos, Music, Movie Libraries are stored on.

If the new Mac mini has the same Storage space as the old iMac..?

Then all you need to do is connect the old iMac's 2T TM backup to the new Mac mini and Migrate from it.

Transfer to a new Mac with Migration Assistant - Apple Support

If not..?

Then you will need to manually move part or all of your data to the 4TB SSD in the Satechi Hub.

If you want to use Migration Assistant to migrate your user and system data to the new Mac Mini (you'll need to have as much storage in the Mini as you have in your old Mac or MA won't work. In that case the Target Disk Mode would be your best bet.

You can get USB C-male to USB A-female adaptors from Amazon.com:

What you are trying to do is not clear..?

You can only Migrate to a drive that has macOS installed on it.

If you want to run the Mac mini from SSD in the Satechi ?

You first need to format and install macOS on the SSD in the Satechi.

Then startup from the SSD in the Satechi and migrate from the old Mac during the initial setup.

If you want to use the SSD in the Satechi for something else..?

Please describe exactly what that is.