Arteta, De Zerbi, ten Hag and Postecoglou are all better / same tier as him. Emery's good but but getting heavily overrated now, let's say Villa finish 6th or 7th, no one will claim he's the 3rd best in the lague.
Not really. He essentially matches the achievements of all of them combined.
Arteta - his best achievement is a second league place and an FA Cup. Emery has a couple of domestic cups and a league win, albeit in a weaker league.
de Zerbi - his biggest achievement is playing good football. Emery's Villa is playing as good if not better football.
ten Hag - won 3 leagues in a weak league with the top dog, and reached semis of UCL with Ajax. Emery won a title in a weak league (but much stronger than Holland's) with the top dog, but also won 4 Europa Leagues. Furthermore, reached semis of UCL with Villareal.
Postecoglou - won leagues in farmer's league with the top dog. So did Emery.
So, while ten Hag and Postecoglou won weak leagues with the biggest team there, so did Emery. Unlike them, he also won 4 Europa Leagues. Like ten Hag, reached semis of UCL with a team that has no business reaching that stage. Like Arteta and de Zerbi, played good football in England.
Achievements-wise, he is easily the third best in the league and likely top 10 in the world.