In my experience, rice quality has a pretty substantial amount of variation, but the cooking device has little to do with it.
While I'm sure that most rice cookers on the market from Japanese firms are optimized and tested for short-grain, japonica rice, I've never had issues cooking basmati or jasmine rice in one, and I've even used them for farro and mixed grains.
I doubt that the rice cooker has much to do with it. I've made rice in heavy cast-iron enamelware on the stove, in a cheap Southeast Asian electric steamer without any fancy "fuzzy logic" electronics, in a fuzzy logic rice cooker, and an mid-range IH rice cooker, which is what we use at home now, and the quality of the rice and its age has a greater impact than the cooking method. I would say that the IH rice cooker produces superior results over our old fuzzy logic cooker, but it's certainly not an order-of-magnitude kind of difference.
Assuming you're located in the US, my benchmark go-to rice brand is "Tamaki Gold", which is from a japonica strain called koshihikari, and I think most of this brand's rice is grown near Sacramento, CA. It's more expensive than the typical Botan or Niko Niko brand calrose rice that's ubiquitious in Japanese supermarkets, but I find the quality far superior, and it's still a good value. My wife tends not to appreciate the Niko Niko or Botan calrose rice very much at all, so we don't eat it at home, but probably 80% of Japanese restaurants in the US are using it or a similar product.
Generally, I'd recommend staying away from the absolute cheapest brands, and choose something that's a couple of notches above. We've used imported rice from Akita or wherever and gotten very nice results, but the differences were far subtler than the price (on the order of $10 vs. $35 for a similar quantity). The sweet spot for quality is near the median price, assuming you're in a shop that offers a wide variety of options.
Edit: Sorry, until your edit, I didn't realize you were working from parboiled or converted rice, I assumed raw rice, as I've never heard of anyone cooking the quick cooking rice products in a rice cooker. (Rice cookers often have their own "quick mode" which shortens cooking time with normal raw rice at the cost of a slightly reduced textural quality). Considering that's what you were starting from, consider using other supermarket brands of rice that meet your target grain size and stickiness/fluffiness, but aren't marketed for speed or convenience.
I surveyed many recipes available by googling "Loation sticky rice", "Thai sticky rice" and "Viet sticky rice".
None of them mentioned kneading anything in after 30 or 40 recipes, although many mention fluffing, folding, or stirring the rice after cooking or steaming to manage the texture.
Several did mention that it is appropriate for eating with one's hands. The key may be the variety of rice that is used. Short grained glutinous rice will have the sticky texture desired.
Best Answer
This lady is pleased with the results just using her one-setting cooker: Sticky Rice. Note, she says in the comments that she uses 2 cups (or slightly less) water to 1 cup of rice (which makes sense) not 3 cups rice to 2 cups water like it sounds like she is saying in the video. She rinses the rice well, but does not soak it.
EDIT: With things like this there are often "camps" of differing opinion. I doubt that Michelin Starred restaurants that include sticky rice on their menu ever use rice makers to make said rice. I am reminded of this: Turning regular noodles into no-boil noodles. Even though there are thousands of recipes online for making lasagna without boiling the noodles, I say pfftthhfft. Sure, you can do it, if you don't mind High School cafeteria food. (Remember the Church Lady? "Isn't that special?" Think like that.)
I am less of a connoisseur of rice than of baked pasta. That being the case, I'm not sure that I would notice the deficiencies of sticky rice made in a rice cooker. With that in mind, and assuming that you are not on a different stratosphere of connoisseur than I am, I recommend that you give it a shot and let us know how it goes.
Yet Another EDIT - I Tried It
I used this brand of sweet (glutinous) rice:
I used 2 cups of rice, thoroughly rinsed, 1 tsp salt, 3 2/3 cups water, and the only setting my simple, old rice cooker has.
My lid is glass, and I could see that there was still quite a bit of water on the top of the rice after the cycle was complete (short, 30 minutes or less), so I waited 15 minutes before I lifted the lid.
Since you can't see the rice without opening the cooker, I recommend that you do the same, wait 15 minutes before opening.
After 15 minutes I checked it out:
It's absolutely fine. Is it as good as the the great Thai place down the street? No. But it's close. The only thing I would (and will) do differently next time is to only use 3 1/2 cups water to 2 cups rinsed rice and 1/2 TBS salt.
After it finished last night, I was suddenly too tired to mess with it anymore, so I just left the cooker on warm, and went to sleep. 5 hours later, if anything, it's improved.
It was fine without it, but I'm intrigued by @GdD's answer here. Later, I'm going to try making the rice again with the above measurements plus 1.5 TBS sugar, just to see (more just seems like too much). Making sticky rice without rice cooker
Final EDIT I Promise:
Yep, that worked. It does brown a bit at the bottom of the rice cooker, but I like that. I enjoyed that as much as sticky rice from the Thai joint. It's not the same, but it's really good. So, there is my recommendation. Have fun!
The lady who made the video has responded to me on YouTube. She seems grateful and flattered. She had no corrections.