I like the feel of cock-on-opening actions better myself. I suspect many others are the same, and I suspect this is all it takes to explain their dominance in the market today. Witness the many shooters who spent time and money converting their perfectly useable model 1917s from cock on close to cock on opening. There are also the theoretical reasons of safety described already in this thread, and the fact that when you close the bolt on a cock-on-opening design it’s far less disruptive to your sight picture, which really lends the cock-on-opening a lot of useability in a civilian context.
Theoretically, if you have a sticky case, a cock-on-opening action might make it harder to open the bolt, because you have to apply enough force to cock the striker as well as to initiate primary extraction. Equally theoretically, a cock-on-closing action is supposed to be a little faster to operate, when the number of rounds fired in a minute might matter greatly. The argument goes that you need to push the bolt closed anyhow, so you may as well make that movement do double duty by pushing a bit harder to cock the gun. So, in a military context, when things like action speed and out-of-spec cases are big considerations, cock-on-closing appears to bring something to the table, which is why it was favored by some countries.
The Brits liked it for its speed advantage. The Japanese adopted it for the same reasons. Paul Mauser used it for his earlier designs, then decided that cock-on-opening was better for his magnum opus, the 1898 action. The US took great pains to make our clone of the ‘93 Mauser, a cock-on-opening.
In practice the Lee-Enfield is it’s own animal, and is the only gun that really has a decisive speed advantage with a cock-on-closing design, in my opinion. All the rest (like the Mauser designs that form the basis for most of our modern designs) require more deliberate operating motions which really negate most of the advantages that a cock on closing action might provide.
In other words, handle an SMLE and you’ll too be singing the praises of the cock on closing design. Then handle a 93 Mauser or Arisaka, and you’ll feel like the action moves at its own pace and you aren’t gaining as much. The Arisaka particularly is a bear to close which makes it feel like your muzzle is sweeping all over creation as you muscle that bolt home.