Salt will work and was used some before herbicides. To the best of my knowledge it was never done other than by home gardeners. The cost is higher than herbicides. There are pluses and minuses for both depending mostly on your feeling toward herbicides.
Bill. Oops. Misspelled the herbicide D-I-R-U-O-N
. To the blackboard 500 times!! Yes there are several herbicides approved for use. 24D is even approved during the harvest and I have done so. I will do it after a frost since even a light one will kill the tender spears. Two pickings are lost. At other times the application will lose two pickings or at least by me. Most spears will twist if hit with 24D and are unsaleable anyway (I won't eat them) I destroy all visible spears for three days. I will stick with diruon at least for now.
Midmo. Roundup will kill asparagus. These are old roots and hard to kill. I would not treat a new patch this way. The spraying is done immediately, generally within the hour, of the final picking. I do a knock down using the lawn mower with the deck all the way down but not running. The spears will snap off while the weeds go under and are sprayed right behind the mower. Any spray touching the spears just breaking ground won't do more than make the spear grow funny.
This spear was probably hit. `
Buzzard Wing. Mulch is the nice way to go. As you state it needs adding to often. The decay adds plant food too. I have 1700 plants and no easy way to get enough mulch or I would use it. I use it on my garden. We grow and sell a small amount of asparagus. Next year is the last year I am going to do this. Getting too old to reach the ground. Or maybe too old to get back up.