"So a relationship with their father is inessential for you then, hooby is it ? Nice"
In essence, no. That's why loads of widows and widowers manage to raise children despite the absence of a father/child relationship. That, however, wasn't the original point I was making. What I was doing was indicating that speaking to the father every time he called, seeing him every contact opportunity is inessential to the formation of a strong relationship with him.
This is a crap comparison, I know, but I wonder of the forces wives, how many of their kids have other plans when dad comes home, how many are engrossed in an Xbox game or not in the mood for talking but still have a firm child/father bond.
I still say that forcing the children will cause argument, discord and resentment. Especially resentment.
I still think that this isn't the business of the OP, particularly not the little digs about how the mother parents during her weekends (mother daring to consider playing out or kids going to GPs normal, for example).
I still think that the DC will pick up on the adult disagreement, not to mention the OP's obvious dislike of their mother and that alone will make them not want to visit their father.
I'm not totally convinced by the assertion that 'offer of trip to theme with mum park trumps bike ride with dad'. Kids will do the oddest, dullest things with people they actually want to spend time with. Maybe their reluctance has nothing to do with their mother. Maybe it has nothing to do with their father. Maybe there are other areas of the father/child and mother/child relationship which the OP considers "certainly is my business" and maybe it's that which the children are happy to avoid. A pp has a good point about kids wanting to please everyone, so maybe, "Actually, dad's girlfriend we don't like you getting involved/playing mum/your attitude to mum" isn't something they'd be likely to say to her or their father.
I'm just musing, sorry, long post!