Try Something Else:
As you've seen, there's a number of people who have provided answers and are generally opposed to the idea of changing from FTE ("full time employment", 40 hours per week) to PTE (part time), due to the number of significant disadvantages that employers generally face.
So, if the idea you had in mind doesn't look promising, try something else.
Other roads:
There is a road, that man people have successfully taken, which may be able to offer you much of what you want. There may be a number of risks, and many people have decided to take this road and have spectacularly failed. Whether you are successful may partially depend on decisions made by people other than you. So this may seem to offer a number of downsides, at least potentially. However, this has a significant upside compared to your suggestion of becoming a part time employee. The significant upside is that many people have done this, with the successful blessing of the people who were writing paychecks at the start of the process.
What you do is you stop being an hourly employee.
Stop being Hourly:
There are two ways to do this. One is to become salaried. Your job is then to take care of tasks, not to submit a certain number of hours. Then, if you get your tasks done early, it becomes your professional discretion to decide whether you remain for any purpose, or take off early or stay at home.
Note: This approach is fraught with one specific danger: many places expect that salaried staff work a certain number of hours per week, like "at least 40". A lot of salaried people don't get overtime pay, and so they end up working more without getting paid extra money for the specific extra time that they work (but they get a higher pay overall, so that cost may be offset by a benefit).
Stop being an employee:
The other approach is... stop being part of their staff altogether. Instead, become a contractor. Bam! You're off their payroll. Now they no longer need to justify those pesky 8 hours of underutilized work. They don't need to worry about whether you qualify for any benefits that you are being offered because you are an FTE. You're just not seen as a drain on management's resources.
Of course, you can't just convert to contractor status and still keep getting paid by the same people just because you think you would like the situation better - your current employer needs to buy into the idea. This may require some salesmanship in your part.
You also get a number of drawbacks. For instance, you'll need to learn how to handle billing. Some such tasks may be something your can outsource. You may feel like your income is based on a less stable foundation, which may not be as desirable in the light of planning to support a larger family. Even if you can get the employer to agree, this might not be a good idea for you. You may not end up reducing the number of roadblocks that you get, but may just be trading one set of roadblocks for another.
Still, this is a decision that many people have implemented, successfully. Some people find that attempting this results in surprises and, overall, things just not not working out, and so some people end up having a lower quality of life after they pursue this.
However, there are also a large number of people who have pursued this successfully.
That's it!
Those are my ideas. Following are some additional perspectives... thoughts that further describe those ideas. However, I've already shared with your the ideas that are meant to be the core of the answer. You may want to consider whether either of those roads seem feasible.
A Success Story:
What you're seeking can be done. I provide you with a specific example. My father was called in by his boss, who said: "We love you. However, we are experiencing some financial hardships. We just can't afford to keep you." My father provided a counter-proposal: "Pay me the same hourly rate, and I will work fewer hours."
The employer deemed this to be a brilliant idea that the employer simply hadn't considered. Now, instead of reluctantly firing someone that the company didn't want to lose, my father's readily available solution led to him salvaging his employment.
The reason this happened to everyone's satisfaction, so well, is that the solution ended up meeting everyone's needs. At the time, my father ended up having five employers, simultaneously! The way he pulled this off has a lot to do with what he did for a job.
His primary role was as a salesperson. So, as long as he could sell some products, he was accomplishing the desired goal. The company cared more about getting the goal accomplished, rather than hour many hours were spent doing that. (And he sold things for multiple employers.)
His other role was a truck driver. (Yeah, stereotypical salesperson and stereotypical truck driver may seen very different. Well, he made a fortune earlier in life by selling, and specifically selling successful deliveries to trucking companies, often as a "dispatcher". So, those two worlds merged.) So if he drove a truck on a couple of weekends, that helped a small company.
The key is that, depending on what needs he served, he was able to identify specific tasks that he accomplished that didn't require him to remain at the office the whole time. If you spend a lot of time in the office, you may not be able to simply convert your scenario into something like his. But my father was once an office worker, too. What he did was identify what specific tasks were being done, and he figured out how he could accomplish those tasks without being stuck to a desk.
A Road For The Future:
It may be that you'll have an easier time doing some of this later in life. One reason is that after having many more years of experience, after you've developed a career, you may have some better intuitive insight into what makes a business happy. That additional business savvy may lead to you being able to better negotiate.
The other thing is that a lot of people have a basic attitude that 20 year olds are meant to serve, under the direction of 50 year olds. Letting a 20 year old punk get away with skipping the societal demand of serving is an idea that, quite frankly, seems offensively unfair to a lot of 50 year olds who already put in their time when they needed to work less glamorous jobs when they were 20. Such people may feel a lot more comfortable negotiating with someone in their thirties who has gained some experience developing their career, as they feel like anything they give up in negotiations is an act of giving something to a good person who deserves such a reward.
I did not just explain that reasoning because I wanted to justify such discrimination of the youthful employees. What I am saying, though, is that such attitudes are rather widespread, to the point that there is a societal trap that forces the extremely vast majority of youthful workers into needing to succumb to the desires of their elders who control much of society's resources.
As you mention your "(soon to be) family", I speculate you may be on the younger side of this story. There is one part of this situation that can be seen as a good side for you. That is, if you fail today, that doesn't mean that you will never be able to accomplish your goal. You just may not have as easy of a time doing this in the younger years of your career. However, in the later part of your career, this may be something you can still dream of, and shoot for.
