When I asked you to do the backup by ethernet.. I forgot how easy that statement is to make but how difficult and potentially expensive it is in reality.
Apple do sell a thunderbolt 3 to ethernet adapter for a $29 in US.
https://www.apple.com/shop/product/MD463LL/A/thunderbolt-to-gigabit-ethernet-adapter
You might find buying a proper dock like the OWC model better value as it gives you a whole range of ports and charging as well.
OR
It might really be better to rethink your whole network.
To reset the TC I need to first save somewhere else 1,5 GB of pictures, movies, music and documents
Actually you don't in terms of a reset. No reset of the TC erases the hard disk. So just to be clear what I proposed will change access to the files.. you will need to do some renaming around the network but it will not delete anything.
However, if you have all those pictures movies music and documents on a Gen4 TC without a backup.. that backup is urgent. The TC is well past its use by date. They can die suddenly and irrevocably without least sign.. except perhaps a backup failing after working fine for ages.. hint hint. And getting your files out of a dead TC can be a rather expensive process. If the hard disk dies.. very very expensive without any guarantee of success.
Just note that I am repairing a Gen4 TC right now for someone who lost all their music. This resulted in psychological trauma and manic depression. And this is far from the first one I have done. He is lucky.. the disk is OK and I recovered the files for him. Do you feel.. LUCKY??
Buy a USB drive today.. and backup your music and files.. do it.. do it NOW!! 4TB for desktop 3.5" type and 2TB for laptop 2.5" type are sweet spots since they do not use archive drives and are cheap/cheapest for capacity.
There could be an issue with your ISP supplied router not handling bonjour correctly. There is a complicated fix.. but let me first suggest you try the following.
#1 set a static IP on your TC.
This is simple..
Open the airport utility and click on the icon of the TC.. click on edit in the summary that comes up.

Go to the Internet tab.
It will look something like this with IP from your main router.

Select Static as your method of connection.

This will swap all the current values without you needing to do anything. And it will be locked.

#2 In Finder use Go / Connect to Server to mount the TC hard disk by IP address.

A menu will popup which you will type as follows.. replacing my static IP with yours.

You may need to give it a password and save that in your keychain.. do it..
#3 you now have your TC hard disk mounted in finder but using an IP that never changes.. and no bonjour requirement to find it.
In Time Machine Preferences click on Add and Remove Hard Disk.

You will see the existing TC hard disk listed by the name of the TC.. and you will also see the same disk listed by IP address.
Add this IP address one.. and use it for backup.. don't delete the original if you are asked .. just say use both.
Tell me if this improves things at all.