Good work, so far, Dick
However, the tab or page in your BIOS I am thinking about is one where there is an "if, then" sequence, there may be a part you have not found yet.
The idea is that your computer will say, effectively, to itself:
"If there is a USB device, then try it first".
and
"If there is No USB device, then try the next one in line, which is the Toshiba HDD"
This list I refer to might also include an optical drive, such as CD/DVD drive.
So after the laptop looks at the T5 and finds there is no OS on it, it should next try the Toshiba, and then boot to your Linux.
You see, the idea we are looking at, in blowing away your OS and then using Timeshift to restore it, hinges upon our being able to access both of the USB install stick, AND the T5, where the Timeshift snapshot is stored. So both need to be plugged in at the same time.
But yes, your "My guess..." could be right on the money, so perhaps restore to the way it was, with the F5/F6 actions, then power down the machine, insert the stick and go back onto the BIOS and see.
In your own time, and well-rested.
Wiz