Howdy
Active Cowboy Action Shooter here. Yes, 38 Special has become more common these days as a pistol round than 45 Colt. A variety of reasons, mostly cost and recoil.
Your question was specifically about 357 Cowboy loads compared to 45 Colt. I don't know anybody who buys 357 Cowboy loads. All the 38 shooters I know use 38 Special cowboy loads. If you buy 357 Cowboy loads, they will by definition be legal for CAS, ie lead bullets and muzzle velocity below 1000 fps. However everybody I know who uses a 357Mag/38Sp revolver in CAS loads it with 38s. When I shag brass I will sometimes find some 357s on the ground. There are a few who hand load 357 Mag down to CAS specs, in order to avoid carbon ring buildup in the chambers, but that is easy enough to scrub out every few matches or so.
Yes, a 357 Mag SAA replica will weigh more than the same model and barrel length chambered for 45 Colt because of the size of the holes. Most of my CAS revolvers are chambered for 45 Colt, but I have a couple of New Vaqueros chambered for 357 Mag/38 SP and they are indeed heavy. It is up to the individual shooter whether or not he likes the balance of the heavier gun.
Yes, I know plenty of CAS shooters who use the term 'mouse fart'. Generally it is only used among friends with gentle ribbing about how light somebody's loads are. If one is not careful, it can be taken as an insult.
Yes, my 45 Colt loads are stuffed to the gills with Black Powder and they are not mouse fart loads.