I would try to look at these options in terms other than "which is better and which is worse?" because either veneer or solid can be done well or badly.

Cherry is relatively stable when compared to many cabinet woods, but yes, any slab door of solid wood will be prone to warp in certain situations, and battens are a typical way of minimizing such problems. Veneered slabs are only as stable as their substrates. Plywood substrates can also warp. MDF is a more stable substrate, but many people equate it (inappropriately) with particleboard and have bad associations with all highly processed wood products.

Quartersawn solid wood is more stable than flatsawn, but in veneer the difference is entirely cosmetic.

Rotary-cut veneers are not inherently bad unless you dislike the look. They are, however, cheaper, so the question of their value gets to be a chicken-and-egg sort of thing: people get wrapped up in the idea that they're cheaper because they are lower-quality, when in fact they're perceived to be lower-quality because they're common, because their cheaper. Rotary-cut veneers inherently repeat their patterns as the log goes round and round; whether this is perceived as tedious and boring or pleasantly rhythmic is largely personal.

Something to consider is that solid wood doors facilitate comfortably eased, or even rounded, corners and edges. Lower-end veneer doors have very thin edgebanding, so the corners can't be rounded because doing so would expose the substrate. I dislike these sharp-edged veneer doors because they're harsh to live with, and also more fragile. It's entirely possible to make veneered doors with thick edgebanding that facilitates a bit of rounding over, but you may have to hunt around to find a company that builds this way.

My preference would be for a better-quality veneered door with thick veneers and eased edges, but I would go for solid wood before I chose cheap veneer work. Really, though: don't discount personal preference. I suggest you worry a little less about these generalities and go to see some actual cabinets.