I've studied a variety of arts, but mainly an old style of taekwondo / tangsoodo (~20 years), hapkido (3), and kyokushin karate (6). The only techniques I've seen in TKD that are specifically intended for hitting opponents on the ground are a few stomping kicks - and they tend to be mixed in with step-sparring exercises and appear in a couple patterns, but not be used in sparring (i.e. read as "we don't practice them enough to use them reliably in chaotic situations"). People also sometimes practice using turning/roundhouse kicks, front kicks, side kicks, pick kicks from the ground to another grounded opponent - sometimes in a downward direction - but again in pre-arranged sequences and demonstrations.
In the older forms of taekwondo - closer to the predominantly Shotokan origins of the art - the ball of the foot was generally used, rather than the instep. I don't recall any mention of kicking with the toes in General Choi's encyclopaedia, though I haven't looked at it for a few years, and I've never seen anyone do it or suggest it in training. Whether you use the ball or toes makes little difference to the chance of having the leg grabbed, and both have relatively small surface area.
I'd say the general attitude / strategy in TKD is that if you've knocked someone down, you expect to have done it well enough that they stay there, and if they don't - when they get up you knock them down again. If you see you need to do more while they're on the ground, you wing it - but there isn't much training for it. Given the increased likelihood of getting caught up and/or taken down yourself, being on your feet and mobile isn't such a bad idea, but it does potentially give them a reprieve and second chance. If you're skillfull it won't be much of a chance though.
It's common to kick with the toes in some Okinawan Uechi-Ryu karate ( see https://youtu.be/HWOn0ZY4ev8?t=44 ), and there's a lot of tough conditioning done to make sure that's practical. I'm not sure if they do anything specific to practice kicks to opponents on the ground though. The whole toe-kick thing hasn't been brought across to mainland Japan in Shotokan or Kyokushin.
Hapkido mixes in more kicks with the takedowns and throws, and you do practice kicks from the ground and to the ground in sparring (if your school spars - not all do) but it definitely suffers from the jack-of-all-trades-master-of-none problem. Often sweeps, joint lock takedowns and throws and are done in such a way to set the opponent up for a kick, or to hold them prone on the ground. If the opponent is moving freely on the ground though, it's kind of up to each person to get the hang of it, rather than there being particular kicks or strategies I've heard enumerated for that situation.