The question posed is "Can a bore snake damage the rifle bore?".
The answer is yes. The potential for damage is there any time you clean the bore, regardless of what you use, even a one-piece rod can make a mess with poor technique.
Is it likely though? I doubt it, but it does happen that occasionally a new rifle is damaged by over-zealous cleaning.
A generalisation, but I think you're more likely to find a bore snake to be the primary cleaning equipment of the "less enthuiastic" shooter, somebody who shoots less ammo in a year than the cost of a nice $70 rod. Others of us buy one-piece rods in various thicknesses (I have one .17", one .20", and several .22" and .30" rods, plus a much thicker gun rod), trying to push patches with a .17 rod down a .243 is a sure way to bend the rod. They have roller-bearing handles to allow the rod to rotate freely with the rifling, and some have brass muzzle guides as well. I also have several multi-piece rods, the short lengths are useful for scrubbing the chamber. I have several bore snakes, including the Otis Cable, all unused, but reassuring to know I have something with me if needed. My field-cleaning kit is the M16 kit that is very compact and clips onto a belt or pack, also unused other than pushing out an occasional sqibbed bullet. Over the top? Very likely, but it works for me
On the other hand I doubt I've ever bought anything for cleaning cars and bikes as I never clean them, I care for my firearms much better than my vehicles.
I have Rose's father's "cleaning rod" here, a length of fence wire he used for seventy years, I really must post some pics of it. Needless to say, his .22 bore is in the most atrocious state I think I've ever seen
The very best "patterns" I managed were around five-inches at 25m. I pushed jacketed .224" bullets through the bore, they "floated" freely through the middle 21", and barely engraved in the 2" at each end of the barrel. Remarkably, jacketed .224" bullets did shoot quite well through it, from memory around 35mm at 25m (my memory was pretty good - it shot a 34mm group at 25m with the 35gn NTX on a Cyclone charged case).
I have since removed the Remington 510 barrel to replace it with a Lithgow Model 12 barrel that I know shoots very well.