I put them in a pan with some oil and some water, bring to a boil, let the water boil off, then fry to crispen up the outside.

I'm pretty sure this is the instructions on all the packages I've looked at.

The ones I buy tell you to steam then then fry, so I'm not sure what you're going on about

I do mine like the instructions say and they always come out perfect.

The traditional (Japanese) way is to fry then steam. I don't know whether restaurants often do otherwise.

I've done both and fry-then-steam is not as sticky, and the water deglazes the pan, if you cook them with anything.

Pierogi though? Boil then fry, they stick like the dickens either way, my husband is the one who makes them - always with kielbasa and sautees them in the sausage pan after boiling them.

I think you're running into a brand specific thing. The last box of frozen gyoza we got said to warm/steam then crisp in a pan.

I'm trying to think and remember if the ones my best friend used to get had the same instructions. I think so, and I'm remembering the smell of them hitting the pan, which wouldn't be the case if they were frozen still. I'm 85%+ confident that brand said to warm then pan fry, though it may have been the brand that wanted you to put them in and get the edges crisp and then add water cover to steam.

But I've seen both processes on different brands for sure.

I tend to prefer starting off with direct heat to get the edges crispy, then steam. Usually, my the time they're finished, enough of the water has escaped the pan so that the edges get nice and extra crispy while the body of the dumpling wrapper gets this elastic and soft texture that lets you take multiple bites without it falling apart, which often happens doing it the other way around.

Hell, I'm okay with only steaming them tbh. I enjoy the soft edges just as much.

But, to answer your title question, I think that it's less about efficiency than it is making sure they get cooked through properly. When you steam/nuke first, you're more likely to try and rush to fry and end up with cold centers. Steaming in the pan doesn't lose much edge crisp (when heat is maintained well), but you're almost guaranteed to have an even temperature throughout. Frozen foods have some unusual processes because reheating or cooking from frozen can radically shift outcomes.

That's why you used to have frozen dinners with weird instructions for the different segments of the tray, like leaving one section covered, another slitted, maybe stopping to stir the dessert partway through, that kind of thing. Dumplings are fairly simple to make fresh because you're actually cooking them. Most of the frozen ones are par cooked before freezing, which means you end up with wonky results if you follow the same procedures as fresh.

Edit: awesome question btw