With Cozy being sent to Tartarus, I beleive, much like with Starlight, it’s due to a lack of proper integration with the world building and story being presented that caused a disconnect with a lot of fans.
So with Tirek, prior to his arrival, season 2 has already Tartarus as, to quote “[a place where] all the ancient evil creatures that have been imprisoned [and that they could] destroy Equestria!” so it’s already been established in universe as existing. Then add on in Twilight’s Kingdom that it’s mentioned Tirek escaped from Tartarus during the events of It’s About Time, so it’s then explained where he comes from. So then when the final showdown happens and Tirek is sent back to Tartarus, the story and world has already been established as the place he was banished and as to why it happened as such.
With Cozy, we don’t get any of that really. Cozy is presented as a normal everyday filly, physically speaking and who’s best trait is at most a silver tongue, but no real threat level she can cause without outside information or coercion from others, so once she’s defeated and outed, her threat no longer exists.
But Cozy is also the only villain the heroes have ever captured and so need to deal with her but there’s been no actual prison or correctional facilities established in universe so literally the only option is the pits of Tartarus, which is not set up as a correctional facility in the episode nor as someplace they regularly use as a prison outside Tirek, so it comes out of left field, nevermind that it seems far harsher than anything the characters have ever done when the likes of Starlight and Discord get reformed and Chrysalis is just able to fly away. And because it was the only option it now feels like there is a very black and white mentality going on with any ethical concerns being shoved aside by going “Cozy was evil, therefore any punishment is completely justified”.
So for a decent amount of fans, the punishment seems over the top (because no other alternative was ever presented in the universe), harsh (Because Cozy was still bad and something needed to be done with her) and seemingly out of nowhere as an option (no pony or literally any other villain besides Tirek, who was already establish to be sentences to Tartarus, has ever been sent to) all for a powerless child.