I don't advise you to get as close as wrestling if you want to defend yourself. Boxing is definitly up to the task, but it can't hurt to know stuff about legs, hence the (kick).
Thing is, fights don't always follow the recommended guidelines. Striking is all well and good so long as you can maintain the correct distance and have enough space to work with. Sometimes, though, that's just not how the fight goes, or even how it starts. Having a good base in grappling techniques is really a good idea so as to not turn into a dead fish the moment someone grabs you.
I used to do taekwondo, where precisely Fuckall% of the curriculum is dedicated to grappling... I believe there was a whole singular once during my years working with the stuff when we had a sparring session with grappling.
I got paired up with a big lump who apparently had some background in school wrestling, so my win on a technicality was rather unexpected, to say the least...
But yeah, taekwondo is one of those martial arts that I tend to refer to as "more art than martial"... Very pretty, not very applicable.
We also had some hand-to-hand in the army, but that was under the direction of a fresh-baked lieutenant who was horribly intimidated by the prospect of teaching a bunch of bargain bin grunts, so most of the classes for the entire week (yes, a whole week!) of combat training were terribly disjointed and uncoordinated.
Sparring is always fun though.