There probably isn't a single group of people more willing to spend time/effort/money in order to complete a project perfectly than hotrodders. The product these people use on metal to cure or prevent rust is POR-15. The rustier the part the better this stuff works. Call it One And Done. Not that stuff you buy at the local hardware store that you have to use over and over, year after year. It's not cheap, but if you truly don't want to have to keep using a product time and again, make the purchase... and be done!
Yes, ALL the paint
on the frame needs to be removed. I don't know of any product that soaks through paint
(as a rust preventative). Rust starts where paint has chipped and migrates under the paint. Ever noticed paint that's bubbled... that's rust.
For those that travel roads, especially during the winter and salts are used, or live or travel near oceans and other salt water ... POR-15 is a very good solution.
There are other very good products available. It just depends on how much work you want to put into it, what you're willing to spend and how often you are willing to redo the work. Isn't it nice to have choices