The performance boost provided by compiling for your specific CPU is real but not terribly large (<10% in the last tests I saw some years ago). Probably not noticeable on common arches unless you're running CPU-intensive software frequently.
Feature selection has some knockon effects. Tailored features mean that you don't have to install masses of libraries for features you don't want, which come with their own bugs and security issues. The number of vulnerabilities added and the amount of disk space chewed up usually isn't large for any one library, but once you're talking about a hundred or more, it does add up.
Occasionally, feature selection prevents mutually contradictory features from fighting each other—for instance, a custom kernel that doesn't include the nouveau drivers isn't going to have nouveau fighting the proprietary nvidia drivers for command of the system's video card, as happened to an acquaintance of mine who was running Ubuntu (I ended up coaching her through blacklisting nouveau). These cases are very rare, however.
Disabling features may allow software to run on rare or very new architectures where some libraries aren't available, or aren't available yet. This is more interesting for up-and-coming arches like riscv than dying ones like mips, but it holds for both.
One specific pro-compile case I can think of that involves neither features nor optimization is that of aseprite, a pixel graphics program. The last time I checked, it had a rather strange licensing setup that made compiling it yourself the best choice for obtaining it legally.
(Gentoo user, so I build everything myself. Except rust. That one isn't worth the effort.)