I think language is better learned by real contact with real language than through a quasi-mathematical approach with drills. It seems to me that mistakes are normal in learning, and there is no need to stress students with artificial exercises to master. Personally, I never got a lot out of such drills when I learned foreign languages.
If the students are gradually brought into contact with the irregularities of language (whether verbs or other elements), I think they do just as well or better than if a whole batch of irregularities are dropped on them at once with the injunction to "Learn this!"
Ideally, an acquaintance with certain adjective forms in conversational settings (well done, thoughtful, lost dog, newly fallen snow, a poorly fed parakeet kept hidden in a stolen cage!), originally learned by rote, can set up the desired "ho-hum" reaction to past participles like done, thought, lost, etc. (I think that sometimes teachers are more panicked by irregularities than students, and teacher panic spreads to the students -- especially once the difficulties are officially recognized in the form of special drills! "Whoa! This is SO difficult we have to spend hours slaving away at it!" 
)
Other irregularities, preferably related by semantics rather than by grammar or morphology, can also be learned by rote at first: came and went, bought it, brought it home, wore it once, and took it back (or threw it out), had (s.o.) over - came over - drove over, sat down and stood up, sat down and shut up, stood up and spoke out, fell down and got (back) up, put it on and took it off, made a choice - chose, had a meeting - met, held a sale - sold, took the lead - led, took a drive - drove, got some sleep - slept, had a fight - fought, etc. There are any number of dialogs you can construct that emphasize the use of such forms, and any number of topics for free conversation which will naturally lead to the use of such forms.
Note that many irregular past participles occur after got. This might be a factor you can use to your advantage in presenting some of the irregular forms. got hurt, burnt, struck by lightening, stung by a bee, stuck in traffic, bitten by a dog, left behind, paid, ...
Students who "want to understand" irregular verbs probably just want to see them grouped in some logical way -- either because it helps them to memorize them that way, or purely out of intellectual curiosity. Just showing them the three basic groups is often enough: Those with t or d endings in both the past and the past participle; those with ng, nk, or g endings; those with -en past participles.
But I think what you really want is better exercises to drill these verbs, that is, exercises that more efficiently drum in the forms and consistently "reduce student error rates", and I'm afraid I can't help you with all that "learning technology" stuff. The ones you've found seem to do the job, and I'm not aware of any "magic bullet" that addresses that goal perfectly.
CJ