chueh, were are you located?
I'm in Georgia and grow sweet potatoes. I purchased a few specialty varieties this year, but I didn't buy any for foliage but for the potato. Their foliage wasn't very tasty.
I've been simply using "grocery store" (probably Georgia Jets) for foliage and it is the tastiest to me, but cooked. I don't care for it raw. I have a link below for descriptions of sweet potato, they mention types with good leaves to eat.
As for rooting a sweet potato. I take the potato, suspend it over water in a vase, with toothpicks holding it up. Let it sprout. Take the small rooted sprouts off of the side of the potato carefully and pot up. I usually grow mine on in a pot for at least 3 weeks. Then 2 weeks after the last frost date (or later, we have warm weather here) I plant those. I just dug mine today, so it takes me about 5 months or so to get a decent crop. 5 months of HOT weather, mind you.
What are your growing conditions like?

Here is a link that might be useful: Sand Hill Sweet Potato

I am not so sure about your exact questions. I don't know anything about Ipomoea batatas, nor "herbaceous perennial". But I know all sweet potatoes have edible leaves, and they are equally and similarly tasteful. I don't know of any sweet potatoes that grow only edible leaves, given long enough time they will (even onamental ones) produce tubers, although they produce bigger roots in hotter climates. Sweet potatoes are not perennial, unless you are in deep south.

Ipomea batata is the cultivated sweet potato. I believe in asian regions there are cultivars that are bred primarily for the foliage eating quality and quantity, which is probably what the OP is referring to. The lack of tuber production could be a result of that, or lack of sufficient growing season.

Thank you all. Yes, I referred to the Asian cultivars that are bred primarily for foliage eating. This kind of foliage LOOKS thinner and softer, whereas that of the tuber cultivars looks firmer.. Anyway.. Great nutrients in leaves. All sweet potato leaves are edible, yet the tuber cultivars can be tougher....

Last year, I did the same, by planting the cuttings. I did not know that sweet potatoes are herbaceous perennials in South right before I posted this thread here. I did pull the roots up last year when it was dying back, because I thought that all vegetables are annuals except asparagus. I am a little south of Atlanta GA. So... I just did not wait long enough for the tuber to form before I pulled it up.

I just found out that Duck Creek Farms (google it) has a variety they say is grown for greens and was given to them so they don't know what kind it is. Might be worth checking it out?