I play with my kids. They are silly. Sometimes I regret encouraging it, because they can be so irreverent and silly as to drive me nuts. Despite this, I have no problem projecting DAD. I'm their father/parent; this is my role. When DAD needs to assert that role, DAD comes out. Kids comply. No choice.
He can't project DAD. I can be silly and have fun. I know how to stop it when things go too far. He doesn't. I have had to step in and stop it for him because he and the kids end up bickering like children rather than him stepping in.
I've observed that sometimes it seems like she doesn't know how to handle the kids. Both of her parents were distant, or not there in their own ways. Maybe it's similar with your kids' father.
That is definitely the case with ex. I think that is why he was so great when the kids were younger. There weren't so many of them and it was easier to navigate. The oldest has commented that dad is great when it is just the two of him. He talks too much about himself. She has commented that dad gets too easily overwhelmed when it is all 4 of them.
It sounds like your kids have picked up on some ideal of how a father should be. You know here's limited; likewise, they seem to have picked up on it, too. Maybe it's disappointing to them. That's OK; it is what it is. Given that, how do you think you can talk to them without alienating them?
It is very disappointing to them, especially the older ones. A lot of the ideal of how a father should be came from how he used to be when they were little. It was much easier for him to diffuse things when they were little. He could get out his puppet and the puppet seemed to instantly diffuse the situation. His old tricks were great for toddlers. Not so much for older kids. Some of it isn't even about an ideal father. It is about having adults that one can trust. They see how I am and how other adults in their lives are and they wonder why dad can't be the same way. We can all be very silly and fun loving and the kids know that they can count on us to figure out whatever may come at them. It is much more comfortable if you are with somebody that you know isn't going to panic or get overwhelmed when something goes wrong. It feels a lot safer to be with somebody that you know is going to remain calm and can deal with any situation that may arise. Dad isn't dangerous yet they don't feel safe with him because they know that he gets easily overwhelmed. In some cases, the kids are better able to handle stuff than he is.
The kids and I have had some good discussions about how to spend time with dad and have it be positive. One comment was to not expect anything from him. Another was to think of him as they would think of a friend. They want me around when dad comes to visit. I try to stay in my room or on the periphery and that seems to help them all relax. If dad gets overwhelmed, I can step in and take over. That isn't an ideal solution because I would like to be able to go off and do my own thing from time to time.
Ex has pretty much handed over all of the decision making to me so it is up to me to find positive ways for him and the kids to spend time together.