Your rational numbers argument does indeed fail, but for a different reason. It's not valid to go from a sum over the rationals to an integral in the reals in t

Your rational numbers argument does indeed fail, but for a different reason. It's not valid to go from a sum over the rationals to an integral in the reals in that way, however it is perfectly valid to go from a sum over the integers to an integral over the reals (in certain situations), if you only want an estimate (see eg. the Euler Maclaurin formula https://en.wikipedia.org/wiki/Euler–Maclaurin_formula https://en.wikipedia.org/wiki/Euler–Maclaurin_formula ). This was fine in Feynman's argument.