I don't shave with dirty razors. It's so simple to remove the blade after each shave and run it under hot water along the the razor or components and dry. Blades are just patted dry and the razor is dried, both with a small microfiber towel.
Rinsing the razor in the sink during the shave us not a big deal. Still, why get "gunked" up water all over the razor? This appears to be a hold over from the early days where there was no running water. Just like when people used to pour water from a pitcher into a sink to wash their face.
I also clean my razor after each use. Why put it in dirty water that you are just going to have to use more time and water afterward to clean up the razor?
It only takes a brief bit of water to clear the razor. Less water than to fill a sink or maybe the same. I don't think that anyone would be wasting a significant amount of water either way?
If I am still going to wash the razor when finished, then why bother filling the sink to shave? You will spend more time and use more water to clean out the sink!
I tend to stay away from using hot water on knives and razors. Unnecessary expansion and contraction of the metal is hard on the edge.