Can I bury the stems of tomatillos the same way I do tomatoes (when I transplant)? They seem to get leggier than anything else, and very delicate...

Nope. Like peppers, burying the stems extra deep won't make new roots, as in tomatoes. But it can help with water conservation and extra anchoring.
Tomatillos may fool you into thinking they are delicate, tender creatures. But in three months, they are like bamboo - tough!

I have seen tomatillos produce roots just above the soil level in my garden. So, I assume that they also produce new roots when I bury the stems. I'm not sure that they produce roots at the rate that tomatoes do, but I do think they do produce some roots from their stems.

Of course, I could be wrong. But, as Loamlump says, burying the stems certainly won't hurt.

Here's another tomatillo tip, FTF - I don't even grow tomatillos from seed anymore, because they come up as weeds in my garden! One year I noticed this, so I pulled up 3 of them, planted them alongside the 4 that I had started (and, as you noted, were leggy, and very easy to break, so I always grew way more than I needed), and observed them for the season. The "weeds" grew better and produced more than the seedlings I started! Since then I have used the weeds and had great crops every season. There are always some tomatillos that get overlooked and over-ripen and fall off, so you just have to be observant and dig a few up.

Dave

I had the same experience with tomatillos. I'm giving groundcherries a go this year, which are close cousin to tomatillo.
I was reading a fascinating blurp on heirloom husk 'tomatoes', which are not in the true tomato family, but in cultivation about the same length of time. These are potentially dangerous plants - all parts other than the fruit are toxic to most people, as well as unripe groundcherries.
Some people are violently reactive to even the ripest fruit - gastric problems mostly, but the rare person has actually went into anaphylactic shock!