However, when I was looking at the UK government website, it seems like you have to make the financial requirement for each step in this process. My understanding was that the financial requirement in the first step was supposed to make sure you were financially responsible enough to live in the UK for 5 years without needing public money, and after 5 years you could apply for indefinite leave to remain. So would I still have to have 62500 pounds in savings to be eligible for both "Apply to Remain with Family" and "Indefinite Leave to Remain"?
Yes, you have to meet the same financial requirement for each of the 3 visas. So, if you are going to rely on savings, you'll need to keep that £62,500 available for the whole 5 years (or at least make sure it's back up to £62,500 at least 6 months before each application).
The £62,500 is actually only 2.5 years' worth of savings... so you need to show it for the first 2.5 years, then again for the second 2.5 years, and then finally for the 2.5 years after ILR.
The figure comes from the fact that the annual income requirement you have to show is £18,600. The first £16,000 of the savings do not count (because that's the amount of savings above which you don't qualify for benefits), then you ALSO need on top of that 2.5 years x £18,600.
So, you have £16,000 + (2.5 years x £18,600) = £62,500.
Hopefully we both find jobs soon after moving, but if we live off of savings for a while, would I be in danger of not being able to renew my visa?
Potentially - but it depends on your financial situation. You could live of some of the savings, but if you weren't able to get jobs, you would need to make sure you could top the saving back up to £62,500 in time for the next visa.
Once you have the first visa though, you can combine incomes to meet the requirements for the FLR(M) and ILR... so you would only need to be earning £9,300 per year each to meet the requirements.
However, we do have someone on the forum who used £62,500 in savings for the first visa, and then used that money to purchase a house, and now they are in danger of not being able to meet the income requirement for their FLR(M) because their employment income might not quite be enough.