IMO bus routes are a key part of the strategy of the transport system. Where the buses go, or don't, is fundamental to the users and so their elected representatives absolutely should take an interest in route planning. In the school analogy this is equivalent to a decision as to whether or not a Spanish course should be offered in the curriculum.
I think the analogy there is a bit off. The actual
routes are not fundamental to the users. What is fundamental is the connectivity provided.
So to me what the politicians are determining is effectively an SLA (service level agreement) - my example above of "nearest stop within 500m, direct service to main local centre, max one change with no more than 10 minutes' wait to central Manchester" would be something like that. But, using the Ashton example (looks at map), whether you do that by way of running a bus from Smallshaw to Ashton and then all the way into Manchester up the A635, or whether you instead do that by providing a bus service that runs from Smallshaw via Ashton town centre to the frontage of Ashton station timed to arrive 5 minutes before all the trains to Manchester (remember it will be one ticket regardless) is really a more operational (though still low-level strategic) decision better made by the PTE.
So in your case the SLA is "we must provide the ability to do a Spanish GCSE to all our pupils if they wish to do so", and that's a "political" decision (by the Head and other governance of the overall school, perhaps). Choosing whether you achieve that by employing a Spanish teacher and having Spanish lessons every Monday and Friday from 10 to 11am, or whether you do it by bussing your pupils to another partner school in your academy group because only one or two want to take Spanish GCSE, is really an "operational" decision for the "PTE" (or languages department at the school).
Or back to the bus argument, what the politicians need to be saying is not "please route the 123 back via Ashton", it's "people living in the Ashton area need to have a public transport service to Ashton town centre at least hourly from 0700 to 2300, your change to route 123 has removed that for people on X road, can you please follow that SLA" and leaving it to the PTE to decide how it does that as long as it does. It might be that routeing the 123 back via Ashton is the way to do it, or it might be that something else can more sensibly be done.
To use another non-Manchester example (as I'm not as familiar with Manchester local routes) the destinations I tend to go to by bus are central MK and Bletchley and those two railway stations. Do I care that the 4 runs via Springfield? No, not really, so if they remove it and put one that goes a different way instead I don't mind, provided the basic connectivity I need is there, and something else serves Springfield instead.