I'm increasingly thinking that TIL don't have permission to agree settlements - that their role is specifically to prepare cases for prosecution
That's not my understanding. I'm happy to be corrected, but my understanding is that the relationship is more like that of a debt collection agency, i.e. TIL get to keep a significant proportion of any money they recover.
It isn't that TIL can't negotiate reasonable settlements, it is that their financial incentive is to rinse people for as much as they can. And the threat of prosecution, no matter how unwarranted, will naturally encourage high settlements; even if you win at the Magistrates' Court, your financial loss from travel/a day off work/other will often be more than the settlement. But thanks to the Byelaws, it is difficult to win at Court unless you did, in fact, have a valid ticket or were given permission to travel. There's no defence of lack of intent or mistaken belief. The fact that the Penalty Fares scheme exists to resolve mistakes appears to be by the by.
I am aware that lawyers and TOC prosecutors working in this field do argue that TIL's charges to customers are reasonable for the work they do, but I am somewhat less than convinced when you see Legal Aid Agency rates for similar work.
The issue, as with private parking enforcement (clamping) back in the day, is that there is no oversight of the likes of TIL, and nowhere to complain to. That is why Arriva use TIL so joyfully, even where a Penalty Fare could have been charged (as has repeatedly happened at Leeds). The solution can only be legislative (like when clamping was banned in 2012) but there is a cost to that which DfT won't want to pay.
And I know TIL read this forum (something posters seeking advice need to remember!) so, if they can publicly correct anything I've said, I will gladly retract anything that is inaccurate.