If that is the case, why did BT fight so hard to retain Openreach? I'm sorry, I am not convinced that BT do not influence their wholly owned subsidiary company. I know that certain curbs have been put in place in the last few weeks, but only under screaming protest from BT.
There is no day to day influence over Openreach. The rules have been in place since OFCOM insisted on the separation, which was when Openreach was formed several (10ish) years ago. The recent talk has been about nothing really, just the regular regulatory review. Its only been picked up by the media because the woman heading up Talk Talk has been especially vocal, as probably looking to deflect their huge drop in customers following their free-for-all access to all customer details snafu last year.
Apart from the tiny minority, Openreach are a requirement for all ISPs, BT included. BT know that, and what to keep it healthy, and more coverage. I suspect that we'd have near universal (ie, all cabs and E/O lines, though some subscribers would still be too far from their DSLAM) Superfast coverage if BT controlled Openreach... ...which would be better for everyone (all Openreach reliant ISPs and their customers). But OFCOM want it run as a separate entity, so any cabinet upgrade has to be commercially viable to Openreach, and can't rely on extra revenues that BT may earn (by being able to sell additional services).
In the highly competitive ISP market we have in the UK, Openreach don't make a huge amount of money against their investments, certainly not enough to do R&D for the next generation of internet (Ultrafast, and beyond). This is one of the few areas where the larger BT Group can help - it can do the R&D. BT has been at the forefront of the next step to enable Ultrafast in developing and testing g.fast.
If Openreach were to be forcibly sold off, rather than the legal separation currently in place, BT would end up with it's most important supplier not being able to provide what BT wanted. BT's future strategy involves a need for fast, universal internet connectivity for its customers. Openreach's meagre profits would end up entirely swallowed by both its own pension deficits and its invester's who would need a decent return, so no further R&D, not further expansion of the Superfast/Ultrafast/Whatever Next networks. There would not be another supplier willing to fill the gap outside of the major, more profitable areas. That's why BT are so keen to ensure Openreach's health. Its also why Openreach are so keen to remain part of BT Group.
But if you think BT Consumer (the bit you deal with) have any more influence/contact/control over Openreach than Talk Talk, Sky, Zen, A&A, or any other UK ISP then I'm afraid you are incorrect. They all have the same suite of systems to use. No more, no less.