Oooh, it’s neat to hear you can get away with harvesting the seed pods young and green and finish maturing them indoors like that. I have wondered if that would be a viable technique.
Being able to harvest a higher quantity of seeds in any given year sounds like a wonderful boon to any breeding goal! 
I believe Cultivariable has mentioned in several places that root crops tend to become sterile over time if they’re replanted without ever needing to set seeds, in general. I can’t remember if it was him or somebody else who said that might be because of long-term genetic damage that just happens over time (perhaps by an accumulation of viruses). In any case, the vast majority of root crops that are cloned for hundreds of years without their seeds being planted seem to become sterile (or almost entirely sterile). I’m not sure exactly what the mechanism is, but it may be something along the lines of senescence – maybe plants experience a decline in fertility when they hit old age, much like animals do.
I think it could also be that if a plant is putting energy into vegetative reproduction it is not going to then waste extra energy on producing seed.
When this happens over multiple generations the plant will begin to focus only on vegetative reproduction, because that is what it has had success doing. That switch is likely a chemical/hormonal change.
I’m sure there are also ways to trigger flower/seed production again. Maybe through decreasing ability to reproduce vegetatively by being root-bound, or other stress.
That could be, but I believe a decrease in fertility usually happens over the course of decades to the same cloned genotype that was originally fertile, so . . . it seems like there’s more than just genetic shifts over generations going on.
It seems like it’s probably something that was either in the genes all along (such as senescence) or an external factor (such as a buildup of viruses).
A buildup of undesirable mutations may be another explanation, because the cells do go through many generations. It just seems unlikely that random mutations would affect fertility so much and affect vigor and yield so little. But it’s possible it affects everything across the board, and we just don’t notice because it’s so gradual.
As for triggering fertility, Cultivariable has said in a few places that stress can trigger flowering and seed set in crops that are usually reluctant to set seeds, so . . . I find that very promising! Especially since drought stress is something I can easily supply to a crop that isn’t happy about dry hot climates, ha ha. 
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If plants experience senescence, just like animals, I have to assume it happens in line with the expected lifespan of the species, just like with animals. I mean, a two-year-old mouse is ancient, whereas a two-year-old human is a toddler.
So perhaps a 200-year-old oak tree may consider itself middle-aged (they can live up to around 300 years, right?), whereas a 200-year-old sweet potato may consider itself to be in extreme old age. Maybe? It’s just my own personal thought, but it seems a reasonable hypothesis. I wonder if it would be possible to test it and get a definite answer? I wonder if somebody already has?
One strong argument against that hypothesis is that coppicing trees seems to reset their biological clocks, and that has been used to make them “immortal.” It’s possible root crops reset their biological clocks every time they’re cloned.
But . . . I don’t know, that doesn’t seem very likely to me. Trees don’t expect to lose their whole tops on a regular basis, while root crops (that are native to temperate climates and have stems that die off in the winter) do. It seems to me the effect of coppicing to reset the biological clock is probably related to severe shock, and I can’t see temperate root crops considering it a shock that winter comes every year, so I don’t think it would reset a biological clock.
Of course, that’s assuming they have a biological clock, which is a huge assumption to make. They may not.
Anyway, it’s an interesting theory!
I think we’d all be interested in hearing about the ways to trigger flower/seed production again that you’ve accidentally discovered. 
It’s possible the techniques you’ve read about and discovered will only work for sweet potatoes, but it’s possible it may also work for other species that are tricky to get any seeds from because they’re uncommon root crops that have been cloned for centuries. Information like that may be valuable for someone who’s dying to try breeding mauka, for example.
No worries. Take your time. I’ve already learned a lot just from what you’ve already shared, thank you!