Awww... I know how excited you were and how much you've done/are doing for the little guys so I'm going to share probably waaaaay too much information with you about how I raised my quail and some descriptions of their personalities and boundaries in the hopes it helps you connect more with your birds
I can share some methods I've used but it is both nature (genetics) and nurture (experiences) influencing their behavior so it's up to you to watch them closely (but not like a predator hehe) to figure out who to approach and when to make you seem harmless and normal to them, or at least some of them
They're birds and as such they appreciate being talked to - fortunately for my quail, I had already developed some pretty pro sickeningly annoying singsong bird talk so I started the relentless high pitched nonsense chanting while they were hatching in the incubator.
"Hi babies, hi good girls, what're you doing?? What good babies, hello!! Oh hi little babies! Mhmmm!!!"
Haha! But really, it makes them comfortable - they can recognize you and your welcoming and friendly "song" - think about the opposite, approaching them in silence, huge and hovering above, watching them straight on with binocular predator eyes while they fantasize about how you'll crush them like a bug :O hehe!
With extra spooked animals of all kinds, it's good to sit sideways to them and not give them direct eye contact but rather do an activity by yourself calmly while mostly ignoring them (can say a few annoying people bird songs now and then lol) until they're more comfortable with your presence.
It's possible they're associating your visits with panic because they panic when you come and they keep panicking until you leave then calm down, justifying to their mass self that the thing to do when people are around is to panic until you're gone, then they're "safe".
You can teach them they can feel safe in your presence if you don't bug them til they relax and you will see some loosening up first - resuming normal behaviors such as preening their feathers, eating, stretching, etc. and these are likely the same quail that aren't as jumpy as the rest when you approach their cage/are working on it
As for helping them get over the fear of being touched, only the very tame ones don't mind head touching - as you can imagine when you compare the mating ritual it's natural for them to buck at it.
I do not use treats to tame quail but it's a good idea *as long as you aren't pushy* - you should be nonchalant and offer it to them *after they've calmed down in your presence* and eventually just leave it for them without chasing them down/shoving it in their faces - this will create a positive experience with you associated with the treat without them even having to contact you.
As for desensitizing human touch - sit with them, watch them - hopefully you can tell some apart and watch them in particular. Yawn, stretch, scratch, shuffle your feet, all while happily and calmly talking to them until they resume normal behavior. This will signal they accept your presence and aren't as likely to flinch when you approach.
You can change their water, watching for the calmer ones to drink first and appear relaxed and concerned with normal quail stuff. When they take a drink, you can risk one light pet if they don't flinch from your oncoming hand. Don't rush things - that will only set you back.
I've found the best way to desensitize them is to do this - pet the back of a *calm/distracted* quail lightly with two fingers and then stop before it realizes what you did and runs away in fear. Ideally, after you pet it, it should keep doing normal behavior.
In this way of only one soft stroke on the back or side of a calm/distracted quail, you make human touch happen with no bad or even unusual consequences. You weren't manhandling them or chasing them around with your finger - it just happened and was okay! The other quail see this as well, even if they weren't the ones being pet.
Then you can build on this - the quail should appear more calm in your presence if you change your behavior around them and subsequently their view of you and the "danger" your visits currently bring.
If eventually some quail see your pet and don't react, you can try petting twice then *lay off* to ensure it was a positive experience. Watch them, give them time, be patient and hopefully some quail will start realizing you aren't an evil human

I wouldn't try picking them up for fun, just when necessary to try and keep most experiences with you safe and positive feeling to build more and more trust
I would estimate you will find success in at least several quail while some will still be crazy, believing the hand that feeds them is an evil entity, no matter what you try and do xD
This is my experience, I have some super tame quail and those tips are how they came to be. I find the white ones to be the most skittish but I have a nice male and female in a group of five who approach me and seem to like being pet (but still will fight for their life while being held haha) while the other three white girls are nearly ramming their little bodies into the other side of the cage which is six feet across
I have a golden girl (literally called Golden Girl hehe) who I can cup her whole body in my hands and stroke her head with my thumbs and she just revels in it. She will usually "popcorn" dance after which I verbally encourage a popcorn party with the other quail in the covey (two tuxedos and a beautiful black with red highlight Tibetan). If I pick her up she is usually unsure and starts freaking out
One of the tuxedos is a female (Pingu) and she went through an independent phase but she's back comfortable in my "clutches" while I singsong some bird baby talk that I hope never goes public. She's fine with being held, at least for short distances, never tried anything longer with her.
The male tuxedo is Robby and while he used to be a cuddler, he's nearly done maturing so he just usually bucks and bursts into popcorn happy hopping whenever I try and pet him like before but he's fine with being held hehe!
Then there's the female black/red Tibetan, Cleo, she's okay but doesn't appreciate the "molestation" of feathers I give the others girls so it's a few nice pets for her and sometimes a double hand cupping. That sounds so weird

She's not fancy on being picked up!
And then there's Lap Quail....... I'm trying to integrate her with my pharaoh breeding group (which contains Big Boy, the original Sleeve Quail [he would stick his head in my sleeve and fall asleep in there standing up until I needed my arm again and he would be forced out, panting]) but she isn't having any of it :O
The presence of Big Boy has really ****** her off and that really surprised me considering she was so submissive to the other girls in her group, making that high pitched tweet whenever another girl put her own body over hers, lol.
So with Lap Quail, I would take her and her friend outside on the back deck sometimes in a big Tupperware of sand, only about 6" high. They would hang out with us and never tried to fly away. I talked to them and gave them a half lid for shelter and pet them until they were comfortable enough to dust bathe and have fun

I don't think the majority of quail realize they're in captivity so they don't try and escape
Anyways, thought she would be an amazing addition to the breeding group based on her friendly and gentle demeanor but I was incorrect so she's partitioned in their cage which I moved inside for the winter for fertile eggs.
This means I get to take Lap Quail out and although she likes to futilely dustbathe in my clean carpet, her favorite thing seems to be when I wear a fluffy sweater with long sleeves and reach my hand in and cup her chest in my palm, then she chirp talks and pecks at the sweater at my wrist almost like she's tucking herself in, lol! Then she will usually tuck her head in to her body and go to sleep and eventually outstretch her neck over my wrist and sleep while standing with me holding her chest with her head hanging - so cute and strange at the same time xD
I've picked these quail out from about 100 I hatched in small batches over the summer. I sold some males and females in groups and culled some males (not happy about it but I was aware getting in to this that it would be necessary and they had a good life and a humane death) - as for your question of whether or not blocking the males from the females will work, unfortunately I don't think it will. The males will start mating with each other and it will be an unhappy place, lol. That's best case scenario with worst case being them fighting and hurting each other :/.
So there's my super long post hehe, hope it helps!!