You will also find a more old school train of thought on various military forums that says you don't need a gym, circuits and running are enough. All credit to them, it clearly worked for many many generations of blokes who naturally still stand by that opinion now. What would be interesting would be to see how much higher the failure rate used to be (particularly due to injury) back when the more 'old school' guys went through training. I know when I joined RM training and got quite far through it, it was mainly injury (and the link between lack of fitness and injury) that saw me not succeed. I'd never stepped foot in a gym. Now nearly 10 years later, going for military goals again, and I'm 10x fitter which I strongly think the gym helped me with, and just feel 10x more robust and less likely to be injured because I'm stronger and can take the bumps harder.
This is an interesting topic, but I'll just throw in that I think there is middle ground between these positions.
Certainly, the old PFT style appoach where you just did pressups and situps and ran is not the best style of training. Equally certainly, many just trained for the test, and just did those things (the test was that way because it was easy to do anywhere with no kit, which was a major advantage before well-equipped gyms). All of this was inefficient, and probably contributed to injury etc like you suggest.
That said, it is
possible to do just bodyweight exercises - or with the addition of a backpack with weight in - that do vastly more than that. My wife is a pilates and yoga instructor, and aside from occasionally a block, there is no kit required. Pressups and situps are not the only, or the best, bodyweight exercises. You can develop quite remarkable muscular strength and endurance without any gym equipment (look at the kind of exercises pro climbers or female gymnasts do). Understanding
how to structure your running training is the key component to improving, getting the right balance of effort sessions, adding sprints and hills, and so on. Nobody needs cross-training on other equipment (at least not for running): it's a bonus to achieve an ideal, quicker, not a requirement.
The reason for pointing out that you can do all this with running shoes and bodyweight is, frankly, that is all that some have access too. If you do, great, make best use of it. But nobody should think they can't achieve excellence because they or their family cannot afford, or don't have easy access to, a gym. This isn't a politically correct equality-for-all argument, it's a practical observation: wait until you see the Gurkha recruit platoon, half your size in full kit, motoring across the Catterick training area. No gyms in the villages they came from!
That said, if you're serious about this as a career, can afford and have access to a gym, then it definitely provides you a lot of good options, so it should be at the top of your spending priorities. Once you are in the Army, the gyms are free, so you are only looking at a year or two membership. Even though the headline rates can be quite steep, work out how many nights out on the piss that is to you (usually not many these days), then ask yourself which is more important.