Gothitelle has been primarily used on perish trap teams as it has an (almost) unique ability. Wolfe Glick, The 2016 World Champion, has won regional tournaments with this style of team so it’s clearly not a bad strategy. (He’s also used Gothitelle other times- this was my favorite.)
I’m case you don’t know what perish trap is, this YouTube video explains it pretty well.
Basically, perish trap employs the move perish song and various trapping effects. Your opponent won’t be able to switch out, so you’ll get 2 free KOs if you can stall it out correctly. Then, you can do it again with your back two Pokémon. If one of your Pokémon is the slowest thing on the field, you’ll win. It’s an interesting style as you technically don’t have to do any damage.
In the past, gothitelle was the only good trapper, as wobbuffet is bad and Whirlpool/Fire Spin only target one opponent. However, Mega Gengar was a great perish trap Pokémon, as it had both shadow tag and perish song. That’s obviously gone now, but new things like Sandaconda and Centiskorch’s Gmax Move and Grapploct’s Octolok (Haha, say that 10 times fast) also prevent Pokémon from switching, although Octolok only targets one Pokémon as well. Dugtrio can also trap Pokémon, but generally doesn’t have enough bulk to be effective in perish trap.
So basically, Gothitelle is usually a staple on Perish Trap Teams due to its almost unique ability, but it did have some use competitively even in 2020. Gothitelle won a regional tournament this year and the player mentions in the report that it allows you to pivot into a favorable spot while locking the other player in his position.
Overall, Gothitelle had a niche role as a perish trap staple which has slightly diminished over the years, but can still be very advantageous in getting a good board position.
As to singles, Arena Trap and Shadow Tag are banned in OU because:
Arena Trap removes one of the core aspects of competitive Pokémon—the ability to switch—which makes playing and building around Pokémon that are vulnerable to an Arena Trap user far more difficult.
Obviously this would be the same with Shadow Tag. Source
I’m Ubers, however, it’s not banned, I’d imagine because of more powerful offensive threats able to get rid of it faster. Even still, Gothitelle is up there in usage ranks (around 40th), and is mostly used as a Choice Item + Trick Trapper. (Shadow Tag Gothitelle is also on the top of Showdown’s Featured Replays- The GetBacker Vs CrashingBoomBang one. Obviously, you should never set hazards in front of a Gothitelle as a majority of them will trick a choice item on you.
Overall, Gothitelle had a niche role in doubles as a perish trap staple which has slightly diminished over the years, but can still be very advantageous in getting a good board position. I’m singles, it’s primarily used to get rid of threats to your sweepers to open up the game in your favor, and (Shadow Tag) was so good that it was banned to Ubers.
I hope that this helps and that there aren’t too many links :)