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.

You could easily say exactly the same thing about a single man raising a child.

All I am trying to say is that this isn't about sexism. It's about the difficulty of raising a child and succeeding in your profession.
This is an issue that affects both men and women.
I may not be a woman, as you indicated above, but I am a parent of 3 children and I know exactly what it feels like to have to make choices between career and family.
Please don't assume that men have an easy job of this.

In what way is the following sentence from the original poster supportive or positive? ". It is easier for the guys to progress at work because they get a lot done for them – dinner, laundry, baby-sitting - all they have to do is focus."

Of course Athena Swan isn't making much difference.
That's because what the original poster is describing isn't sexism in the workplace. If it is sexism at all it's sexism in the home.
Athena Swan answers the wrong question and really the lack of a difference should be telling people that.

You just have to be careful that you are not dancing to other peoples tunes.
At the end of the day you have to be passiomate about your research.
In my opinion this should always be the top priority.
People are generally hired because they are good not because of the lab they worked in.

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.

In my opinion you should find universities who support this method and ask them directly.
Some of the advice you are getting on this thread is well meaning but its poor.

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.

If you have a PhD in Computational Chemistry then you will know that computers (whether quantum or not) will always need to be told which problem to solve and to have their results interpreted. In both cases that can only ever be done by a human being. The job of the Computational Chemist for example is to identify a problem in chemistry, make the calculation and then interpret the result in terms of the chemistry involved in the system. For most computational chemistry jobs the software is simply a tool used as a black box. Quantum computers won't change that. In fact it will make more problems solvable which currently are out of reach.

Scientists and engineers have been talking about replacing humans from the workplace since the beginnning of robotics and yet here we are with record employment and a massive increase in the types and complexities of work available. Jobs may change but plentiful, intelligent work will always be available. I believe there is absolutely zero chance of routinely available, affordable functioning quantum computers arriving within the next 5-10 years and absolutely zero chance of the type of AI required to remove jobs altogether being available any time within the next 40 or 50 years in my opinion.
Synthetic Chemistry jobs will go first and computers are a real threat to that.

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.

The bigger concern is that at this stage of your PhD you seem to believe that not being able to reproduce someone's results means you have disproved their results. As explained above, you haven't disproved anything. If you publish that you HAVE you are going to have a major credibility issue in the viva.

If you really want to do this the. you should certainly start talking to potential supervisors.
I would imagine you would find a position.
The bigger concern is the 2:1 regardless of which university you got it from. To be honest, in terms of your education, the university is largely irrelevant. The fact is that you have learned barely 60% of what you have been taught. That knowledge gap is going to be a problem and you will need to close it urgently or the PhD may end up being a nightmare. Spotting links in interpreting your data for example and generating ideas for example will be much more challenging for someone without a 1st class. It can be done though.

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.