That said: speaking as a cofounder of a brand new company, and despite being corrosively skeptical of YC in the past, if it weren't for the SFBA prereq, we'd apply. You're right, it's annoying!
And that's to say nothing of investors, over 70% of whom are concentrated in the valley.
Plus, there's also probably some signal value here for YC. If you're not sufficiently committed to be willing to move somewhere else for 3-4 months, you may not be that serious about your startup. It's not the strongest signal, but its impact is probably non-trivial.
I agree that there's value in being face to face for mentoring like this, but I just don't see why it has to be in the most expensive place to live in the U.S., especially considering the fact that you want to be saving money when starting a business.
Penny pinching about $10k is completely irrelevant if you are hitting the growth rates that ycombinator is looking for.
... Then you might be older than 30. I don't blame YC at all; I'd do it the same way. However, lots of the industry's demands for evidence of passion/interest/excitement are things only applicable to young people.
Consider how many people with similar commitments (kids, marriages, mortgages) get deployed for military purposes every year for roughly the same amount of time. It's manageable for families.
But there are possibly easier solutions to consider. I'd recommend that people with families consider flying back every week or so for a few days. Working remote for a few days and then spending the weekend with the family will be fine, and I've seen lots of people keep that type of arrangement going for many months.
It's very common in the film industry for people with families to work remote on location for most of 12-24 months.
Does YC ask you what you plan on doing after 3-4 months? For example, if you wanted to move your company back to Cleveland after the summer, would they not like that? Would your potential next-round investors not like that? I'd have to imagine they'd much rather have you stay in California.
I don't know what their preferences are on answers, but it's known that most venture capitalists prefer that your company be based within a half hour of their offices, so if you're basing the company in a location without funders, it may be hard to fundraise.