These ass-jet toilets are not intended to replace toilet paper. They are for getting your asshole (and ladyparts, but I can't testify on that topic) much cleaner in much less time than with conventional American toilets and dry paper alone.
You use the ass-jet, then paper. That's how they work. (For a heinous bowel-movement situation, you might do paper, ass-jet, paper again.) You aren't supposed to skip the paper!
Using your mud example, if you got mud all over your head, would you rather just keep scraping your head with dry paper towels, and use up a couple rolls worth and still have some mud left over, or use some water too? Same principle.
If you take a perfect shit (love when that happens), then sure, it just slides out and leaves behind minimal debris. Great. But let's say you ate a plate of Uncle Jim's nachos the night before, along with a twelve pack of PBR, and you definitely didn't achieve shitting perfection this time. That is when these toilets really shine.
The number of times you have to wipe your ass to achieve that comforting pure-white-no-residue final wipe, that tells you your asshole is clean, is astronomically higher with paper only than it is with an ass-jet plus paper. I mean, have you ever had one of those wipe-it-ten-time-and-dammit-it-still-isn't-clean kind of shits? You just never have that happen with the ass-jet. The water helps wash your doody-hole AND that moistens the toilet paper (for the first post-jet wipe). That makes it work better, just like a wet dishrag is more effective at cleaning a dish than a dry one.
I personally would be surprised if populations that lack ass-jet toilets didn't have a higher incidence of hemmhorrhoids from all that wiping, over a lifetime. I don't think we have that long-term data yet.
But from personal experience, they have saved me thousdands and thousands of asshole-wipes over the years, and I could never go back to a the barbaric American toilets of my youth.
(When I moved back to America several years ago, I brought a Toto washlet toilet seat with me. And the new apartment that I just bought in Tokyo had many options to specify, but the toilet wasn't one of them--just as patio11 suggests above, the place came with a brand new whiz-bang model featuring the latest in shitter technology from Toto, complete with not only heated seat, ass-jet, and wall-mounted control panel, but also sensors that allow it to raise the seat automatically as I approach, and flush for me when I am done.)