While FHA does have one of the lowest down payments in the industry it comes at a price. There is up front and monthly PMI or Private mortgage insurance that stays with the loan for quite a while.
Many are getting into homes with that 3.5% down payment only to find the PMI is eating them alive. Last I checked there is up front PMI that can be rolled into the loan, and monthly PMI was .85%, sounds innocent enough but on a $636,000 mortgage amount that might mean an additional $450 per month and that can be an alligator. Especially if you are buying rental property. This additional $450 per month could be like having another unit.
Private Mortgage Insurance isn't to protect you with regards to your mortgage. It is there to protect HUD/FHA against their losses. The lender is guaranteed that their money is safe up to 96.5% therefore they need to collect 3.5% down payment so they are 100% whole in the case of a default.
Most 203k buyers or refinancing clients would refinance once the work is completed. Quite often the same lender that made the loan in the first place will attempt to refinance that loan for you.