IMO you can replace them with something similar but that movement is a common one and needs training.
I personally have a reduced ROM in my shoulder and it makes pushups harder. I also can't bench very much but I have found dumbell presses don't cause me issues, perhaps try them.
I don't think push ups are necessarly better than dumbell presses for the arms/shoulder/chest they are just extremely convenient. However they do require some core strength so perhaps some additional planks (high planks are better) or similar would be wise.
Replacing them will not be a short cut but you can only do what you can do.