How is the remote control working
It might be true, but it could just as well work the other way around too.  Probably real systems do this different ways.

I know that my TV has non-volatile memory in it, since it powers up to the same channel it was shut off with, without the remote being active.

Here is a simple way to test this:

Use the remote to set the TV to channel A.

Then use the remote to go to channel B.

Bring the remote to a different closed room so that it's IR can't get to the TV.

Select channel C on the remote.

Select channel D on the remote.

Bring the remote back to near the TV.

Selet last channel.

If the TV goes to channel C, then the last channel is stored in the remote.  If the TV goes to channel A, then the memory is in the TV.