Many exercises are simply not safe if your muscles are tight or you lack the flexibility to do them properly. Swimming, weight lifting, most sports come immediately to mind.
The accepted wisdom amongst my crowd is that the appropriate warmup prep is some dynamic movement to raise the body temperature, like rowing or biking, followed by practicing a given movement at progressively higher weights, until you are ready for the work sets.
I've never heard a weightlifter recommend static stretching before training.
It took <3 months to fix once I saw a physiotherapist that gave me some dynamic stretches to do in between my workouts. I increased my squats from a paltry 2/3 of my body weight and reduced range of motion to 1.5x my body weight at full range of motion in 6 months from I started doing the stretches, after having been stuck without any improvement for many months. I've kept increasing since.
I don't doubt that doing the movements may work for some, especially if you have reasonable flexibility to start with, but it certainly didn't work for me, or at least did not give noticeable results over a very long period of time, whereas it took less than 2 weeks from I started doing my stretches until I saw substantial improvements.
I absolutely agree with you that pre-exercise stretches is not recommended for lifting weights, but that does not mean stretches are not useful.
Strict good-mornings are also quite good once you have decent amounts of flexibility. Strict = make sure your back stays arched, push your butt as far back as possible, with a slight bend in your knees, and hinge/bend at the hips. Essentially you should feel it in your thighs and butt and not elsewhere; if you start losing the arch in your back, you've pushed as far down as you can go, go back up and repeat. It's more important to keep form than forcing yourself as deep as possible - you'll get better depth soon enough.
I've later incorporated some static stretches including several of the one in the wikihow link above, but the ones above were enough to see dramatic improvement when done a for a few minutes a couple of times a day.
I also do asian squats regularly (so effectively sitting down in a squat position, unloaded, for as long as I an do - a few years back I wouldn't be able to get up without intense pain from that position, if I got into it in the first place...) and feel that helps me increase the depth I get into (and it's made it so much easier to play with my young son, to be able to effortlessly squat down to him instead of ruining my knees...).
Glad to hear dynamic stretches worked for you!
I find that foam rolling, lacrosse ball rolling, and band tension exercises help the most before powerlifting. My knees don't seem to track properly for the squat unless I foam roll my legs and glutes beforehand. It makes a huge difference.
Light static stretching is good for relieving tension and tightness after the workout. But before, foam rolling is much much better for making you flexible without losing force and tightness.
What you absolutely don't want is to stretch before the exercise to loosen up the muscle - you're not suddenly going to gain much extra range of motion, but you will reduce your ability to control the movement.
Stretching after (our outside of) your regular workouts, on the other hand, can be useful.
However, overall flexibility seems to be a longer term condition than just pre-workout stuff. Stretching post-workout or daily but disconnected from pre-strenuous exercise seems to make you more limber/capable of movements that require more flexibility to safely/correctly do. In particular, most of the serious dancers I know stretch(ed) a huge amount (often daily) and also did fairly strenuous dance (often daily), but the static stretching and actual exercise were highly removed from each other. There are also a lot of movements in most types of dance that are either impossible or dangerous without a certain amount of prerequisite flexibility.
Pre-workout warmups are also seem important for avoiding injury: if your muscles are tight/tense and you throw yourself into a strenuous exercise, my experience and everything I've heard says you're going to have a bad time. From personal experience, but every time I've injured/hurt my muscles it was during exercise where I hadn't warmed up or had done a bare minimum of warmup.
Source: Mainly personal experience and what I've seen as a former competitive swimmer (high school varsity, butterfly, fairly serious), former dancer (modern; not serious, but I know some people who were part of larger dance companies), and someone who is currently involved in fairly intense bodyweight strength training (think gymnastic conditioning).