I'm going to go against the flow and state that cheap "ryokan" are, indeed, inauthentic -- and there's in fact a very specific species of cheap budget hotel that labels itself "ryokan" precisely to attract foreign travellers, while being nothing of the sort. Kyoto has some egregious examples, eg. the "Backpackers' Ryokan" Budget Inn and its evil twin the "Capsule Ryokan":
The Japanese definition of a ryokan is actually really simple: it's an inn that serves you dinner in your own room, usually course by course with maids fluttering about in kimonos, and that dinner will be spectacular (and expensive). If the dinner is eaten together in a common dining room, it's a minshuku (民宿), lit. "people's lodging"; and if they don't even offer dinner, then it's a doya (ドヤ) flophouse for day laborers, or its modern incarnation the backpacker hostel. Obviously reality is not quite as sharply delimited as this, but those are the expected definitions of the terms, and seeing a ¥3000/night flophouse calling itself a ryokan is bit like seeing a Motel 8 rebrand itself as the Palace Deluxe Grand Hotel -- the new nameplate won't fool anybody who checks in for long.
The difference between the three may seem small, and you're probably wondering why not just book a cheap place and go out for a nice dinner then, but the idea is that the ryokan itself is the experience: you and yours stay inside from check-in to check-out, enjoying the food, the baths, the art and the tranquility. A minshuku is a more social experience, with people eating and chatting together, and a doya/hostel is just a place to crash for the night.
Also note that I didn't say "tatami" anywhere: while overseas visitors usually equate ryokans with tatami mats, this is not always the case, and an increasing number now have Western-style beds (at least in some rooms) to cater for older people who have trouble sleeping on the floor. For example, here's one of the Hōgetsu rooms at Tagaya, a famous ryokan in Wakura Onsen:
Note the little tatami alcove in the back with cushioned seats and tea equipment. This type of half-and-half wayōshitsu (和洋室), lit. "Japanese and Western room", with Western bedding but some Japanese-style space is also quite common.
All that said, it's a nice only-in-Japan experience to sleep on tatami mats, and so if you can't swing the ¥10,000+ per head it usually costs for the full-on ryokan experience, then a pseudo-ryokan is better than nothing -- but a real ryokan it's not.