You are all looking at this wrong. Chansey is not bad because it makes stall viable - a) stall needed a buff anyway b) stall is still not that viable (the same common pokemon obliterate it - NP Togekiss, NP / SD Mew, SD Heracross).
No, where Chansey starts getting broken is on balanced and bulky offense teams, where it basically serves as a panic button meant to cushion the impact of mispredicts. You play like shit and your let opponent get Nidoking in for free? Just go to Chansey. That same opponent in the same game gets CB Weavile in for free as well? No problem - Chansey handles that just fine too. You're having a terrible day and let an important sweeper get crippled in addition to all this? Fat Whore to the rescue.
The closest thing UU had to something like this before the tier shift was Snorlax, which is very common for exactly this reason - it counters a large number of otherwise extremely dangerous threats while using just one team slot. But Snorlax is much easier to break on both sides than Chansey, it relies on a 30% chance of paralyzing incoming attackers as opposed to a 100% chance, it has no access to instant recovery, it cannot support the team in any way except by being fat and taking hits and (sometimes) phazing. While chansey has its own disadvantages (lack of offensive presence, leftovers) its strengths more than make up for this, and allow it to be the new "check of all trades" in UU. But where Snorlax was balanced by its total inability to take physical hits even from scarfers (at least not without severely compromising its special tanking ability) Chansey can tank ridiculous physical blows and ridiculous special blows while offering far more in terms of team support.
Why is this bad? Because it takes absolutely no skill to slap Chansey on your team and let it save your ass from various mispredicts that allowed your opponent to build offensive pressure. The philosophy of the senate has consistently been to encourage balanced play, and our definition of balance incorporates the very important factor of skill and team diversity. This is Pokemon, and you ought to expect and be prepared to handle a variety of threats without having hard counters for each one. You ought to suffer for poor play without being able to rely on a crutch like Chansey, and you ought to be able to wrest victory from your opponent without having it taken away by something that avoids a 2hko from silly shit like +1 Kingdra Outrage and +6 Cofagrigus' HP Fighting.
We have this tier exactly where we want it - skill-based play revolving around synergy, prediction, and a wide variety of different Pokemon, to where even half of RU is viable in UU on the proper teams. Chansey threatens this balance and diversity, and it should not be allowed to stay.
Everything was beautiful. Nothing hurt.
I'm back, baby. Look for me in 5th gen UU!