I've recently shut down my previous company and am working full time on my new idea.
I currently have ~4 month before I dry out, and am selling one of my cars in order to have a longer run way.
I really think that working full time on your ideas makes you X10(estimated) more productive, since not only do you have more time to work on it, your mind is also empty and totally focused on only one important thing.
Focus is something that people underestimate, but it's sometimes what makes the difference between success or failure.
If you are serious about entrepreneurship, and you can sustain a few month without working part time - my personal advice is that you do it.
I agree with your point of working full time will make you 10x more focused. The other important thing when you are at your very early stage of product development is not to be distracted by other stuff like how you are going to do marketing, scale, etc. etc.