Again not really true,
OR have a price list so they cannot charge service provider 'A' something different for the same product , to provider 'B', what the end user is charged (if they are charged) by their provider isn't set by anyone other than that provider, so a charge by provider A for a inappropriate OR visit my be £60 and another provider may charge their end user £100, please don't reply saying BT would charge £130, it's irrelevant , the fact they SP's their own price , and can 'mark up' the price is the point,
OR have to charge all providers the same, that price is quoted on the OR website, to do anything else wouldn't be treating those company's equally, and OR have to do that (and they do) if they didn't OFCOM could remove their licence .
You say problems are dealt with by OR regardless of provider, well obviously, if you rent something with maintenance included, and if a repair is needed and it's on what OR are responsible for, then the repair is free, but again , this is where the level of service varys depending on the service provider, and gives an end user the chance to compare them, not because the repair would vary , but the way the service provider interacts with OR may.

A cheap no frills provider may be more likely to have customers that if presented with a bill for an inappropriate OR visit, wouldn't pay, but simply move to another provider, but that cheap provider still has to pay OR, so perhaps this cheap provider is less likely to call OR out on their customer behalf, in case they didn't pay them should the need arise.
It could them follow that perhaps a customer of a 'cheap' provider with a genuine OR fault may struggle to get them to call OR out, in case they got stuck with the bill.
Perhaps a cheap provider is more likely to pay it's staff peanuts (and gets monkeys as a result ) so the level of diagnosis when one of their customers contacts them with a problem may leave a lot to be desired.

What if , in the OP case , perhaps if they were charged £60 by EE for a OR call that wasn't warranted, and EE said they wouldn't be charged,but were , another (more expensive) provider would have diagnosed the problem properly or given the end user better guidance that enabled them to solve the problem themselves and save £60 as well
It can be the case that a small extra payment up front saves money in the long run.

Ofcom cant remove the license they own the national network it would close every phone line every where. all ofcom can do is fine them and ultimately make BT sell off OR

your quoting that EE definitely said they would not be charged in any circumstance this is just hearsay

and it is BT OR systems that diagnose the problem NOT EE's EE just have access to the test system and these test MUST be done before BT OR will even attend
if the BT OR system says the fault is on the bt network then there would be no question of charges if the bt network tests ok but there still is a fault then OR would attend on the proviso that a charge would be made if the fault is found not to be on the network.

quoting pages and pages of waffle does not make you quote correct of the insistence that EE definitely said there would be no charges

A no thrills provider gets the same service off OR as anyone else and to say there is a difference (BT retail will give better service because they charge more) is against ofcom rules