I have some sympathy for your DP as I find the system to be completely rigged in the mother's favour in 99% of instances. My step brother went to court numerous times (I think last count was 5) to gain access to his daughter (who's currently only 5) due to a malicious ex.
For people saying surely she's in contempt of the court order - she is, but my step brother found that the police would say it wasn't their job to enforce it, he would have to pay to take her back to court and then the court would just say something along the lines of "you're in contempt of the visitation order - here's a new one" and the whole process would start again. It was a nightmare and he has said he can't afford to keep doing that every few months.
However, although I understand it's expensive and difficult, I struggle to ever support the idea of people giving up on their kids. I hold everyone to the high bar by dad set when my parents got divorced when I was younger - he spent a year almost bankrupting himself fighting in court to make sure I was allowed the choice of where to go (I was 12 so a bit older) to live (and this included fighting a number of allegations) because he knew I was desperate to not stay with my mum.
Unless there is absolutely no money to keep going back to court I don't understand why these dads aren't doing it! As someone said, if she did make false allegations then that would be difficult, to put it lightly, but how is she making all these threats? If he ensures that the only discussions they have are via text/email then he'd have those threats in writing which would blow them out the water in court very quickly.
Courts involve CAFCAS prior to the hearing and on the day and they literally couldn't give a monkeys about fights between parents, they see it all the time and are very good at cutting through it and just making a judgment that they feel is in the best interests of the child. If your DP already has a court order in place, it would be highly unusual for them to not find in his favour again, and the court would question why she didn't bring up the allegations the first time around.
Also, as others have said, there is absolutely no one stopping him from texting/calling her. However it does put him in an awkward situation. I don't believe it's ever in the child's best interest to put them in the middle of these things so any sort of variation at all on "I would love to see you but it's up to DEX" could be problematic and if she asks why he's not seen her then he either has to do that or lie and look like a bit of an arse.
But I would have to find a way around it if I was him, it would break my heart if I couldn't speak to my ds.
But I also would struggle to be with someone who I didn't believe was doing everything in their power to set up access/communication with their children. What do these dads will think happen when they're older? Do they think the kids will just happily accept that the ex wives made it difficult so they decided to wait it out? I wouldn't want anything to do with them if I was a child in that position.