I like the basic idea but have a few comments.
Normally a compost heap should not be under a tree. Tree roots will grow up in it, sucking out nutrients and making it hard to turn or harvest the compost. I made that mistake. If I don’t turn it a lot, it becomes a pain. It’s quite possible, if you don’t let it get too deep, that the chickens will keep it turned enough to defeat the roots. I’d be really tempted to try a compost heap right next to the coop and eliminate the tunnel altogether. You can always do the tunnel later if you deem it necessary. A little trial and error may save you a headache later. Things seldom turn out like you plan anyway so you need to be flexible.
You don’t say where you are so I don’t know how hot it gets but a coop in the shade is usually good. Heat kills more chickens than cold, a lot more.
If you build a tunnel for the chickens to get to the run, you will not be able to get a lawn mower or any equipment over it. You probably won’t be able to step over it either unless you build steps up and down. Since you are probably going to have to walk around it anyway, why not make it tall enough that you can easily walk in it? That might be really convenient for you. Say you use a dropping board to collect pure poop for the compost and extend the life of your bedding. It would be easier to carry that straight there instead of going through doors and gates.
I don’t know you chicken density but unless it is fairly high there is a chance that some weeds and grasses will grow in that tunnel. Chickens don’t necessarily eat every green thing that grows where they can get to it. They eat the things they like and leave the rest when they have a choice. I have a large area enclosed in electric netting that I have to mow a few times a year to cut the unwanted stuff so the things they like can grow. You may find you need to mow or weed eat in there.
Mine really enjoy the compost but remember to put fairly high sides around it. They will scatter it all over with their scratching. If you don’t put sides around it to contain it you won’t get any compost. It will be scattered.