iPhone DOES have emulation options. Delta is one of the best emulators, but it can only do Nintendo (and only up to NDS and N64). We have RetroArch as well, but not all of the cores. Worth looking into.
For Android, I've heard good things about the latest Galaxy A phone. It's around $200 USD, 4GB of RAM, and an Exynos processor that is supposed to be pretty good. Screen goes up to 90Hz, too, though it's only 1080p. But I feel like $200 can buy a better handheld for emulating.