I deleted the time machine backup from external hard drive using finder. Can I get my files back?

I deleted the time machine backup from external hard drive using finder, because I thought I had moved all relevant data over to my new MacBook already. Unfortunately there are some very important personal audio files that I forgot in the data migration process.

The drive has not been written to since then. Theoretically I am sure all the data must still be on there (drive still shows 445 GB data used). The folders have simply been "deleted" so time machine doesn't know what to do with all these unorganised bits on the drive.

When I connect the drive to my MacBook and start the migration assistant, it gets stuck at "looking for other sources". So I suppose the drive is not being recognised as a drive that has a time machine backup stored on it.

I have tried to recover my backup using the easeUS Data recovery Wizard following this guide:

https://www.easeus.com/mac-file-recovery/recover-time-machine-backup.html

By doing this I am able to see many different files, including photos, documents and audio. But unfortunately not the files I am looking for.

The following video shows what the software is able to "find" on my drive:

https://youtu.be/76qxcgywai0

With this software I am able to "recover" those found files and export them to my internal drive, but I think what I really need to do is to recover those files in place, meaning I need them to be recovered in their original location on the external hard drive, not exported to my internal drive. This way (I hope) the disk would be recognised as a time machine backup drive again.

Ideally I would like to get my external drive back to the state that it was in before I deleted this folder.

Since deleting doesn't mean overwriting, I am hopeful that there may be a way to do it.

Any tips on how to proceed from here are greatly appreciated!