I've read that the original mats were tatami on hard floor, and I have to wonder if people also trained outdoors on dirt. There's obviously a greater chance of injury, compared to modern mats. Do modern mats facilitate the teaching, and, if so, why?
Judo has always used mats. In Judo Memoirs of Jigoro Kano, Kano (the founder of judo), measured practice spaces by the number of tatami mats.
The warehouse afforded a practice area of only ten tamami mats, and there were square pillars here and there.
The primary protection against injuries is technique and cooperation, not mats. The beginning lessons of judo focus on the practice of falling (ukemi) and throwing for safety. Falling training aims to either roll out of falls or spread impact over a large area (arms, legs, back/side) and to protect the head, spine, and other joints. Throw training aims to put the partner onto their back, but not the head or the spine. There are no points for throwing someone on their head, and trying to land on your head to avoid giving up a score will disqualify you. In training two partners each try to throw the other, but not in injurious ways. Although mats provide some protection against head and other injuries, if you land on your head with any regularity, you will get badly injured.
Competition judo mats are not plush. There are "crash pads", which are significantly thicker (~15cm) and softer mats that inhibit movement because they are so soft. You can't really have matches on top of them because you sink into them too much while trying to move. Some practictioners like to use crash pads for heavy throwing practice, but crash pads are not necessary and not all facilities have them.
So in my opinion, modern mats have not greatly affected the practice of judo. Judo has always used mats to improve safety, but the primary safety features of judo are inherent in its design and not in mats.
public gymnasiums started to appear in the US in the early 1800’s,Inspired by early health/vitality philosophies of various cultural movements. Eventually, gyms made their way into schools and universities, and even became large independent organizations, like the YMCA and such. Gym mats evolved in the late 1800's;the desire to have surfaces you could fall or land on, and soon wood floors; were replaced by soft surfaces like cotton stuffed pads and mattresses.