I think it's really about the route field of the ticket as a whole. Some Advances don't actually have a TOC restriction, for example "& Connections" as you say, but Southeastern even use "Any Permitted" or "Plus High Speed" for many of their Advances for instance.- &connections tickets (though it's debatable with regard to the main reserved leg)
On the route in question the fares vary between TOCS. So if I buy the cheapest i don't suppose that I could use it on another company's train.I think it's really about the route field of the ticket as a whole. Some Advances don't actually have a TOC restriction, for example "& Connections" as you say, but Southeastern even use "Any Permitted" or "Plus High Speed" for many of their Advances for instance.
Similarly GWR often just set them as "AP X" where X is some station along the intended route they want you to take (e.g. Slough).
I don't think you can be penalised for taking a different TOC to the one your original train is for, if the alternative TOC isn't excluded by your ticket's route. Condition 5.2 of the Advance ticket conditions refers.
On the route in question the fares vary between TOCS. So if I buy the cheapest i don't suppose that I could use it on another company's train.