I'm a dr, but not a surgeon.
Sounds like the OP's husband is a registrar (for non -drs this is someone who is fully qualified as a dr, has done postgraduate exams, and is in the last part of training to become a consultant, which is usually 4-5 years, although can be more if does a PhD or a fellowship to gain extra experience).
Although I am not a surgeon, I am aware that the hardest part of the job is getting the hands on experience of doing surgery- the way that training is set up these days means that people get a lot less experience than they would have done before. There are good reasons for this - we operate far less overnight, and only for life or limb threatening issues, as we know that the days of fairly junior surgeons just being expected to get on with operating on anything that came in, regardless of time of day, wasn't safe and led to patient harm, but it does result in less experience overall, and so most surgeons are desperate to get to theatre for operations (after all, they became surgeons because they enjoy operating).
Again, for important training reasons surgeons are not expected to simply turn up in theatre and be allowed to do the op- they generally need to know the patient's case, which means reasing the notes, familiarising themselves with what's been going on, and meeting/examining the patient, all of which takes time before the actual operation. There is a strong culture/expectation of showing commitment by doing such things, and when you are a registrar, most people want to impress their bosses, as they can potentially be applying for a job in that hospital.
As a registrar, there is much less flexibility compared to being a consultant. It is likely that once a consultant, if they are in a specialty that does on calls, they will have a day free in the week (some of them choose to do private practice in that time, but they may not), and also have the ability to do some admin from home. They can potentially request to do LTFT (less than full time training - note that even LTFT training in medicine means working what most people would consider a FT job - a standard medical training job is approximately 48hrs/week - so an 80% LTFT job will be working 38.4 hours). Traditionally (not just in medicine) it has been unusual for men to go LTFT for childcare reasons, and there is still a worry that by doing so, they can be perceived as "less committed".
A 6am start is likely to be because Op's husband is in a hospital that is some distance away - again, this is not something he can control during training, as they are required to move around different hospitals in the region while training.
It sounds like he is making an effort in terms of taking the kids out at a weekend to spend time with them and to give OP a break, but it also sounds as if the timing is not so good for OP, and OP is not feeling valued or recognised that he is able to do what he does because she is holding the fort.
I'd suggest a good talk, at a time when neither of you are feeling more stressed that usual, about your feelings and the options to manage this. There are lots of things that can be considered - husband going LTFT (but note above comments, or if he is close to finishing traing, he may just want to push through for 1-2 years FT to get there), buying in help such as afterschool care, mother's help to help with the drop offs, cleaner, a list of jobs that are his responsibility in the week, things that he can do at the weekend to prepare for the week and make life easier, or simply a commitment that on one/two evening per week, he will be home (barring TRUE emergencies) at a certain time and do all bath/bedtime etc. In the last situation, it wouldn't be a fixed day, due to on call rotas, but you ought to be able to look at his rota for the next 2-4 weeks and pick out a day from each week that suits.
Also, talk about what he thinks a consultant job might look like - think about compressed hours for example. I know a (male) paediatric cardiologist who does compressed hours to have 1 day in the week free for family time.