There is not much else to say about Dugtrio, so I will try to approach it in a different direction in this post just for fun: What if Dugtrio had Arena Trap in gens 1 and 2?
Dugtrio could use it's excelent speed and crit rate backed up by a strong STAB EQ to trap and potentially kill a lot of things. Dugtrio could switch into a thunderwave or thunderbolt from a below 75% chansey and 2hko it, surviving one Ice beam at full health. Chansey could try spamming Softboiled, but that will probably result on a winning crit from dugtrio (or a fp from a para'd chansey). Dugtrio generally 2hkoes Alakazam, but zam ties in speed with dugtrio so it would be safer to para it first. Being potentially able to take out special walls will definitely make spc sweepers such as starmie and lapras more dangerous and effective, and will contribute to a faster meta in general (also the fact that dugtrio kills but also tends to die fast).
Dugtrio is really close to 2hkoing Golem and Rhydon (or ohko with a crit), although it can only take one eq in return. If you get Dugtrio into a rock-type without taking damage, you will generally at least ensure that zapdos or jolteon can ko it in one hit. This, and the fact that Dugtrio could easily trap Jolteon (so zapdos loses competition for the electric-type and also a minor check) would very likely make zapdos more common.
Lastly, with Arena Trap Dugtrio would also be the premier revenge killer. Due to its high speed and decently powered eq, duggy would be able to finish off most 25-30% things, including the always dangerous taruos and other weakened mons.
Dugtrio would probably get more specific uses in gen2; it'd be more similar to ADV in this sense. The first mon that comes to mind is Raikou. Duggy is 10 points faster than kou, and can 2hko with earthquake. If raikou doesn't talk, just forget about surviving if you've just rested. If kou talks, it would have to hit 2/2 hidden powers, and that's assuming you attack the turn dug comes in. Even if raikou's not sleeping, a healthy duggy would win as long as it doesn't switch into hp. Taking out raikou will open up a lot of stuff, such as zapdos, gengar, misdreavus, jolteon, vaporeon, exeggutor... so it makes for a faster meta obviously.
Apart from raikou there are other eq-weak things that would be susceptible to being trapped by dugtrio, probably becoming more unreliable: Gengar, Nidkoing, Jolteon and Tyranitar are examples, although all can 2hko dugtrio (and duggy falls to 2hko ttar). Dugtrio could also prevent Missy from getting away with perishtrap.
While Arena Trap Dugtrio seemed to be useful for chanseys in rby, in gsc it doesn't do as much to blissey in gsc. EQ hardly 3hkoes Bliss, although Dugtrio could always try Beat up I guess. Or maybe dugtrio could also get away with a curse set against blisseys not using ice beam?
GSC gives Dugtrio the option to run Hidden Power, so it'd probably choose Bug to ko weakened Exeggutors (does 50-59%). There is also the option of hp flying for heracross while sorta covering eggs with sludge bomb, but i'm sure hpbug would prevail as the most common choice.
And lastly, duggy would aslo be useful for revenge killing very low health mons in general, it'd be especially useful for things that can rest (or recover) later on such as snorlax, starmie, vap, very weakened suicune/miltank etc.
So in short, arena trap dugtrio would make for a faster meta in both rby and gsc as everybody was expecting already before reading this!