Divine Retribution
Conquistador de pan
You have to look at the bigger picture. Psyshock with Analytic and Stealth Rock down will do 47 - 53.5% to 252/252+ Chansey and 50.3 - 57.5% to 4/252+ variants. Now Chansey is under a lot of pressure. If it intended to come in and set Stealth Rock, use Heal Bell, or poison/paralyze something, it's going to be taking a huge risk. Starmie can now 2HKO with Psyshock if it doesn't heal up immediately, and the Starmie user gets a free switch to whatever other offensive threat he/she wants if it does heal. Not only that, but a second Psyshock added onto the earlier calculations makes it 73.5 - 85% vs 252/252+ sets and 79.4 - 92% vs 4/252+ sets. Just a little prior damage and Chansey can't even switch consistently into Starmie. It's the basic concept behind Charizard Y + Tyranitar/Bisharp cores. Charizard Y struggles mightily to get past Chansey on its own, but everyone knows that Chansey is still far from a guaranteed answer to Charizard Y because all it takes is one Pursuit (and maybe a little Stealth Rock damage to boot) for it to no longer be a counter.
On a side note, Blissey can run an ADV-esque CM set on certain bulky teams that need a better answer to TG + RD Manaphy, but that set is pretty niche and rarely used anyway.
I'm sorry, but did you just compare Starmie, something with a rather mediocre base 100 Sp. Atk, to Mega Charizard Y, something with 159 base Sp. Atk and Drought?
No, there's a massive difference between the two, and what you've just said confirms that, although maybe you don't realize that. Not only does Chansey need to have some prior damage on it, but Stealth Rock needs to be up, you need to predict the switch and use Psyshock (which has rather poor coverage and is easier to switch other threats into than Charizard's spammable Drought boosted Fire Blasts), and you need two consecutive high rolls in order to beat it. If those stars all line up, then yes, you can get past Chansey, but packing Psyshock for that alone is a waste of a moveslot.
As I already said, Psyshock on Starmie is not primarily for Chansey. It allows him to beat Calm Mind Keldeo, who can 1v1 him otherwise, and also allows him to OHKO Gengar who try to spinblock him and deal solid damage to Mega Venusaur, who fears nothing else in his movepool. These all make Psyshock worthy of consideration, but I maintain to say that simply slapping Psyshock on your Starmie does not mean that your team no longer has issues with Chansey, and nothing you just said has convinced me otherwise.