Real talk, forget about a down payment. There are a bunch of different ways to get a 0 down mortgage with varying qualifiers so that chances you qualify for one of them is quite decent.
Even if not, there are still a bunch of other ways to get low down payment mortgages for ~3% down or less.
Toss out the old adage of “20% down or bust” and keep any money saved towards it for savings for all the other costs of home/closing
That’s great, in theory. In reality, you’ll get stuck in a perpetual savings cycle like OP and in many cases never reach the mythical threshold.
200k savings sounds nice, but if you have to spend 5 years saving and housing prices jump 80, 90, 200% in that time that savings lead gets entirely erased.
You can always play around with your interest rate later on, but you can never change what you paid for the house