I've been going through old forums to try to find the answer, and frankly there isn't much. The only good reason I've found is that, with the prevalence of Sandstorm (and Hail to a degree) in Generation 5, Chansey's inability to hold Leftovers made it too prone to being worn down by passive damage. Blissey, on the other hand, could negate this chip damage with its Leftovers and avoid such a problem. Now that weather teams, especially ones centered around sand, are much less common, Chansey no longer faces many issues with its inability to hold Leftovers and thus outclasses Blissey in almost every single way.
There are many other reasons that players had provided for using Blissey over Chansey, such as its ability to use attacking moves outside of Seismic Toss (debatable, but whatever), act as a win-con with Calm Mind and hold Shed Shell to avoid trapping, as well as its lack of vulnerability to Trick/Switcheroo/Knock Off, but these do not really answer your question as they are still relevant in today's metagame.
As Trachy mentioned, it's quite ironic that Chansey's usage skyrocketed in tandem with Knock Off and trapping methods such as Pursuit, Arena Trap and Shadow Tag (when it wasn't banned). One would think that this would make Blissey more viable as it is less vulnerable to such threats. Guess it just goes to show how outclassed Blissey is by Eviolite Chansey.