Hi OP,
I think it is easy to get caught up on lots of issues here.
I have done the full time mum for a few years and also now I work full time. I could not function at 5am on a weekend day now - there is no way I could use this time constructively with my kids nor could I bear for this time slot to be my only quality time with the kids in the week as it would be a total waste for me - I simply cannot function properly at 5am since going back to work full-time and would probably be in the same position as DH.
Does DH spend time with the kids alone at another time? If not, perhaps this can be incorporated elsewhere in the routine so he has a fighting chance to spend some quality time with less phone temptation? 5am is just not a quality time together time slot.
DH obviously supports you getting up with the children. Perhaps he is struggling in his own way (as we all are in the parenting journey at some point) and just wanting a bit of escapism. I get you asked him not to use his phone so much and you feel let down. Perhaps some compromise to be had here?
I would be very careful about steaming in and taking over as you put it. I did this all the time in the early days out of sheer frustration and it did more damage to the relationship than I could have anticipated. I realised I wasn't making space for another way of doing things, and pushed an agenda of what seemed right to me. This made DH feel useless, and he retreated away.
I get what you are saying about him perhaps not interacting with the kids at all. This isn't really ok and perhaps this is really the thing you are worried about here? I have noticed with many DHs that they don't have the same responses to babies and toddlers as they do to slightly older children. Everyone is different, but I notice this in many cases. Perhaps that might be the part to gently focus on, rather than anything else he may or may not be doing to try and get through this phase of life.
So, are you being unreasonable? Not exactly. It sounds like you have a genuine concern about the kids being ignored when they ask for something. Have you let your frustration take over - possibly?
Remember DH has his own ways of doing things which are very valuable for the kids too. On this occasion maybe it's easier to let it go on the basis it is 2hrs and it is in my view unrealistic to expect quality time to be had at 5am. I totally get the frustration and concern, but there will be other benefits to the kids of having to be independent in those hours.
Perhaps the one thing you need to work on with DH in some way is the being "available" if they ask him something, regardless of what he is doing. However, he may realise if he is too available the kids will want to involve him in their games, which is also not what he probably wants at 5am! Ultimately, I am still learning from my DH how to step back and let my kids be independent, but he had taught me A LOT about that, and I will say there is ever such a fine line between leaving them to be independent and ignoring them.
Perhaps DH needs some additional time with the kids at a time when he has a fighting chance of doing some awesome parenting and spending quality time :)
Wishing you all the best xxx