I second the recommendation for "Li" jujube. Asian pears are not considered low maintenance here because of fireblight, otherwise I'd have them on my landscaping plan. For taste and texture, "Korean Giant" is my favorite asian. "Housi" and "Shinko" are more popular, although "Housi" is more susceptible to fire flight. "Korean Giant" has BIG fruit, and ripens very late (mid to late Sept here), so you might want to be sure that'll work in your area. KG is also a good pollenizer for other varieties -- not all variety combinations make good pairs for pollenation.

the new stem from the bottom may be from the root stock? if so, you need to prune it out and get new growth from above the graft. If everything above the graft rotted out and is dead, you could do your own new graft to the new stem using material from your healthy trees. as to why no fruit, its hard to say without more info, but from what you have said here, you may be giving too much water (since you had rot) The main thing though, you say you bought them and planted them this year? It is not unusual for new trees to have no fruit the first year due to the stress of transplanting. The older trees should be old enough. I have Li and Lang, the Lang starts it's bloom about 10 days after Li but I'm not sure that's your problem.

Yes, wild jujube used for rootstock does not taste as good as the name varieties. Rootstock jujube has small fruits and very sour but sometimes if you are lucky to have it crossed with a name varieties then the fruits taste a little sweeter. It is very easy to graft them. You can use bark graft or cleft graft. Dr Yao has a good video to demo how to do it. https://www.youtube.com/watch?v=fFLwOWe0KQ4. Good Luck.

When this tree is young it does not bear much. My other trees are small, but if sweet is what you're looking for, then you have one of the best tasting varieties and don't really need anything else. It will take about five to seven years before you get a nice amount of fruit. The tree needs to get established and is slow growing. Just protect it from rabbits etc. Having any another variety would help pollination. The fruit quality will get better and in a few years you'll be happy you have this one. Just give it about a gallon of water per day during the hot season.