Time Machine won't mount

Is it possible to retrieve the last backup via Time Machine from it and transfer it to a new external drive?

Not directly, not any more that is. Since macOS Catalina, Time Machine migrated to using APFS and no more "hard links" to files which conspired to make it impossible for us to directly copy or otherwise duplicate one Time Machine backup disk's history to a new (usually, a replacement, which is usually larger capacity) backup disk. It just won't work. Yes I have tried.

The much shorter answer to that question is "no".

Where would I find the Mac's existing backups?

Another good question with a convoluted answer, the shortest version of which is "on the backup disk". Before they arrive at that destination, they exist as "local snapshots" on the source volume — in other words the Mac's very own startup disk — which can actually be used to completely restore that very same source volume. It can be very useful to "undo" recent changes to that startup volume's contents, with the significant and obvious caveat that they cannot be used at all if that startup disk fails or becomes otherwise unavailable.

The other caveat is that "local snapshots" have a lifespan of about 24 hours.

All of that illustrates the value of always having at least one additional separate, redundant Time Machine backup disk. You can have as many as you wish. If you get another one, add it to Time Machine, and after it creates a new backup you can take the original one and erase it. Even if it has failed, sometimes erasing it will result in it being viable again. Sometimes it will keep working for years. Or, it may fail again in a very short period of time. Meanwhile, ask yourself how many viable Time Machine backup disks do you have? If it is again only one, then consider getting another one, for the reasons you now know.

It doesn't take too much imagination to conclude the minimum number of backup drives is three: "Two is one and one is none." Mathematically incorrect but justifiable in practice.

That hopefully addresses your last question "Do I really need to backup on an external drive?" which is yes. Assuming you value the information on your Mac. Not everyone does, but for those who do, everyone's backup needs are different.

Backup disks you can use with Time Machine - Apple Support

Something is wrong with that. It should indicate APFS Volume.

Remove the disk from Time Machine, and add it again.

I seriously suggest renaming the backup disk to something other than "Time Machine". There is nothing wrong with giving a backup disk whatever name makes sense to you, but Time Machine is software. The message shown is essentially "A couldn't back up to A" which may be correct, but doesn't make a lot of sense and is likely to result in your own confusion.

Neither one of those changes will affect that Mac's existing backups.

Having said that I also suggest obtaining and using at least one more Time Machine backup disk, and adding it to Time Machine. The reason is that the apparent disassociation of Time Machine and what was a previously viable backup disk (named Time Machine) may suggest some kind of failure with that disk... named Time Machine...