What effect did Pokemon have on the metagame?
Genesect had some of the most absurd set variety in all of AAA, with up to 7 sets being equally viable depending on the state of the meta. This forced teams to try to cover as many sets as they could, although the most common sets consisted of Tinted Lens, SFLO, and Choice Scarf Regenerator. One core that popped up was DShield Mandibuzz + RegenVest Jirachi, which could cover almost all of the most common Genesect sets (since SFLO sets did not run Flamethrower very often) while being good against many other metagame threats such as Power Construct Zygarde-10% and Tapu Koko. Defensive Pokemon also had to have some way of threatening Genesect. Blissey was forced into Thunder Wave, Mew needed Will-o-Wisp, and Jirachi commonly ran Fire Punch. This had to apply to every single defensive Pokemon on a team, lest Genesect get a Shift Gear up and win on the spot.
In what main roles was Pokemon used?
Genesect had two main roles: a Shift Gear sweeper and a Choice Scarf Regenerator pivot. Between its great offensive stats and outstanding coverage, Shift Gear sets were powerful, versatile, and a major threat to be prepared for. Choice Scarf Regenerator pivots flexed Genesect's solid bulk, unique typing, STAB U-turn, and coverage options to constantly pivot around, slap answers with coverage, and just generally be an unkillable annoyance if played properly.
What caused it to have a significant impact?
The most important this about Genesect is this: it was
impossible to be safe against all possible Genesect sets. This is not an overexagerration; Genesect simply had too many options at its disposal for a single team to say "yeah, I do well vs this" against every set. Even if a team were to somehow dedicate itself to solely beating Genesect sets, it would most likely lose to the many other strong breakers in the tier. It also had insane coverage. Flamethrower, Iron Head, Blizzard...you name it, Genesect had it, at least on the special side of things. In fact, Genesect had a whooping 12 types of moves available at its disposal (technically 11 because its only Rock-type move is Rock Polish), which could be further compounded by strong abilities such as Tinted Lens, Sheer Force, No Guard, and Flash Fire.
How do/did you deal with this Pokemon in AAA?
As stated above, you could not prepare for every Genesect set. It just wasn't possible. However, options did exist to beat some or most sets.
Heatran
Heatran was the best Genesect counter in the tier. Despite Magma Storm's dismal accuracy, it boasted a 4x resistance to both of Genesect's STABs, took at best neutral damage from even the most fringe of Genesect's coverage, and could take advantage of Genesect switching out by laying Stealth Rock or trapping the switch-in. Genesect had ways around it (Flash Fire for Desolate Land Heatran, Primordial Sea for Mold Breaker Heatran), but not only were these looked down upon as more fringe options, Regenerator Heatran ran Body Press and could threaten Genesect either way. Genesect often had to rely on teammates to dispose of Heatran before it could make meaningful progress.
RegenVest Jirachi + Dauntless Shield Mandibuzz
Rising as a metagame pick in the latter half of Gen 8 AAA as a counter to SFLO Tapu Lele, it really took off whenever Pixilate Tapu Koko was discovered and became a great choice on many a team. With its great special bulk, Fire Punch, and U-turn, it could directly threaten most special Genesect sets while negating the damage Choice Scarf sets did to it with Regenerator. However, it could not beat Tinted Lens sets. This is where Mandibuzz came in, as it solidly walled most physical variants of Gensect and had Foul Play to punish it for setting. Barring a strange mixed set with enough Attack to OHKO Jirachi at +1 or a lucky Iron Head flinch, Genesect struggled to get past these cores by itself.
Toxapex
Depending on the ability it ran, Toxapex could take on almost any Genesect set. Prankster was the most common, and this ability allowed it to 1v1 most physical sets, barring Galvanize Extreme Speed sets. Volt Absorb was the second most common, and as Genesect often relied on Thunderbolt or Zap Cannon to hit Water-types, it often blanked special sets, as ones that tried to tech VA Pex with Psychic lost to other threats in the long run. However, the reverse also works; Prankster is near useless against most special and mixed sets and can only remove Genesect's boosts with Prankster Haze, while Volt Absorb can potentially Burn physical Genesect and leave it much weaker, but not only did Guts sets exist to remove this weakness, it was always one Tinted Head Iron Head flinch away from losing.
Some defensive cores
While not all defensive cores could completely prevent Genesect from coming in and sweeping, they could at least punish it for attempting to do so. For example, Blissey could Thunder Wave it, Ferrothorn could set up on it with Iron Defense or also Thunder Wave it, and Mew could Will-o-Wisp physical sets and either dent special sets with a Body Press or pivot out with Volt Switch. These are obviously set-dependant and not true answers, but often, a crippled Genesect was a useless Genesect.