Both your supervisors have told you that your writing is not good enough. Ignore that at your peril. You seem sure that the fact that you are using a new method is at the root of their concern but I get the impression from what you have said that it is the quality of your writing which is the issue. It might be worth stepping back here and making absolutely certain you inderstand why neither wants you to submit right now.

Bringing this back on topic and removing the sexism, the question should be how do you balance career and parenthood?
This is exceptionally difficult and in my experience you have to sacrifice something.
You can still have a good career and be a good parent but if your priority is to leave work when the phone rings to deal with your child it is inevitable that other work colleagues who are more dedicated to their work will gain the promotions and pay rises.
Equally if you commit to work you will miss out on the early years of your childs life.
Having tried both of these approaches I am well aware that neither extreme works.
It's a terrible situation to be in.
The solution? I think having a society which provides equal rights to men and women regarding maternity/paternity would be a good start. Also, access to cheaper childcare would help but in my opinion the single biggest thing we could do is to legislate to force all employers with more than 10 members of staff to provide free wrap around childcare on-site for all parents from 8am to 6pm. This would revolutionise the workplace and be the start of true equality.

Sorry but there is no shortcut to reading the latest literature in your field and choosing for yourself.
Only you can know what is interesting for you.

This is very subjective.
Ask 100 researchers what constitutes very good work and you will get 100 different answers.
The only opinion that matters is your own.
There might be a magazine for your discipline which gives the latest research news.

I hate to say this but at your stage of the PHD you really shouldnt be needing anything other than cursory support.
The whole point of a PHD is to become an independent researcher. Perhaps supervisors should make this clearer to candidates because the vast volume of posts in here from students who fail to understand this basic point and end up suffering depression and anxiety suggests they dont.
I see my supervisor no more than once every 3 months or so and I work on my own ideas and really this should be the norm in my opinion. As for working in a lab on your own, with Youtube these days I would expect you to be able to find someone who has videoed useful info on how to do most stuff.
I appreciate you are struggling. A PHD is just about the hardest thing you will ever do and you should expect to feel overwhelmed.
I think you need to take some time to consider how you are approaching this.

Its a time management issue which is not limited to lab based work. I am doing a computational based PhD and I have the same issue. There is always another calculation to do. It is particularly tough if you are not getting results because you cant relax reading papers when you are worrying about having no results. The problem of having to write your thesis wont go away though so you need to learn to time manage.

I have two supervisors but the second person has no knowledge of what I do. Having said that, I dont really have much dealings with my first supervisor either and have only meet up every few months to report progress. Exactly how I like it.