I am a freelance instructor, and while I am not currently associated with any school, I have been in the past. With both schools I worked for, I set my own schedule. In all cases it is a part time job for me, and therefore is after work and on weekends. But while my students are important to me and I try to bend over backward to help, I also have a family that I like spending time with and other commitments.
So, if my schedule and your desired schedule don't match, it's nothing personal, I'll try to help you find another instructor. If you came to me with a request to fly regularly at 5 AM, I'd tell you sorry, but I'm not interested. Because I'm really not, and as it's a part time job, I don't need the money that badly. Or the hours, for that matter.
Will I stay up until 11 PM during the summer to get in a night flight with a Private student? Of course. Will I stay up until 11 PM to fly regularly 2 or 3 times a week (say for instrument training) with someone who gets off work at 7 or 8 PM? No, sorry.
Maybe I am a unique circumstance, but it seems to me that we all decide what we're willing to do for money and what we're not. For me, I usually won't fly Friday nights, and won't fly on Saturday. Does that prevent me from taking on some students? I'm sure it does, but that's family time and is important to me. But on the contrary, I will fly all day Sunday when many instructors won't. When I meet a new student for the first time, I lay out my schedule. If that works for you, great, if not, then I wish you well. No hard feelings, but if it doesn't work, it doesn't work.
From the other side of the equation. Going in first time as a student I pretty much just assumed that the lions share of instruction happened on the weekends, and just wasn't sure about other times, like evenings and early morning on weekdays. I also didn't know that it is most optimal to fly in the morning or evening for turbulence.
I didn't realize that there are a number of people either without jobs, or night jobs, or whatever so while not the lions share, a decent number of folk actually do take lessons weekdays during the day. I assume their instructors for some reason are available also and don't work normal jobs.
I found out pretty quickly too, CFI's (at least here, through the club I'm in) don't make very much money.
I actually fealt I benefitted from briefings (up to an hour or so) before the flight. I found out that different instructors handle this differently. I expected to pay for the briefing time since it is their time.
As a working man, I just don't have the option of flying on weekdays (except Monday's) normally during business hours. I can fly weekends and Monday's so it was no demand on my part that an instructor be available when I am, but it just plain wouldn't work if my instructor most of the time was not available when I can be. It isn't a complaint, it just doesn't work if we can only find common time once every two weeks I won't make progress.
I'm flying out of Oslo, Norway. One huge plus is the long, long daylight hours in summer. We can fly pretty late mid summer, at some times of summer sun barely goes down, but we have the opposite proble in winter. Mid winter it's only light a few hours mid day,and weather is more unpredictable.
In my case, my first instructor was fantastic as far as availability, and he did the best job I've experienced with briefing before the flight, and debriefing after. He charged for that time too and I was happy with it.
But it turned out his day job flying survey ended up taking more time than he expected and he had to drop instruction. I then got another instructor, but he couldn't fly most weekends, and even if he could he never knew for sure until thurs. or Friday. I also didn't feel like he was thorough enough, or that interested in teaching.
So I had to ask for another. I asked for someone that mostly might be available on weekends.
He is a really excellent pilot, flies acrobatic for fun and competition, was trained in the Air Force here, and for a time I was worried he was too good and maybe expected too much too soon, but revised this and think he is doing it right. But he doesn't take pay for preflight briefing, and seems not to want to spend a lot of time on theory or what we will be doing, how it works, etc. on the ground.
This puts me in s slight dilemma as I really know I benefitted from getting the theory and asking questions, checking my knowledge for mistakes on the ground before getting up there. Different style here.
That said, I totally appreciate him practically donating his time, he really doesn't need the little money that CFIs get and he is generous and loves flying and teaching I think.
He intimidates me a little though. It was a more natural experience for me with the first instructor. It fit me maybe a little better. Sometimes I feel (we are around the same age, but he is expert and I am novice) like I'm back with my dad who could be kind of sharp in tone, and dismissive of my dumb questions.
Anyway, I do thnk the OP is expecting a LOT with wanting 5 a.m. Lessons. Unless of course he finds an instructor that is one of those up early raring to go guys...but I wouldn't expect or really even suggest such early lessons.
I'm very lucky to have the instructor I have, he also seems to be available a lot of the same times I can fly and he is an excellent pilot so I'm happy with this.