I know I will get flak for this especially as this is a new account but I actually felt the need to sign up to answer this one. It is a bit premature in my opinion to say either way if this guy is a creep or (just as you described him) enthusiastic. But on the basis of what we have been told so far for so many of you to cite things like "so many red flags" and for you to state that you are "f*ing furious" to have been woken up by a message seems to me to be a bit of an overreaction. I hear over and over on here about how difficult it is to meet someone as a single parent but I really wonder how anyone expects to keep a relationship going without any tolerance whatsoever.
From what I have read so far on threads about dating on here, if a guy doesn't text frequently enough then he is considered to be ghosting and doesn't have the courage to end things or he is a player who is probably too busy with other women but if he does text then he can be considered a creep because he did it at the wrong time.
Yes 4.00 am is early and if you are a light sleeper and having been woken up can't get back to sleep it could be considered annoying but frankly if I had met someone I was interested enough in to give my number to and chat with until 11.00 pm, I'd be flattered to get a good morning text. And if it did upset me then I'd try to be a little more diplomatic about it.
"Hi xxxxx, what time is it where you are and what are you up to? It's really early here so don't be offended if I don't reply for a bit, I'm going to try to get some more sleep. I'll message you back in a bit when I get up and I'm properly awake."
If what he did really bothers you then explain the time difference, say you need to leave your phone on because of DC and your ageing Father but state that you are a light sleeper and a message or call at 7.00 am (or whatever time) would be great to help you wake up as you need to get ready for work, get the kids to school etc.
Everyone says they want to meet 'normal' people, well normal people make mistakes and as several have said he probably expected you to get the message later when you woke up. If you think it is a red flag that he didn't think before pressing send them by all means move on but if we are talking about red flags, someone who gets "f*ing furious" at being woken up by a text would really make me concerned about what I could expect in the relationship if I did anything else wrong.
FWIW, I'm in a relationship with someone who suffers from anxiety, depression, PTSD and BPD so perhaps my tolerance and patience is a little higher than most but showing a little bit of understanding in life can go a long way.