Imo 5x5 should be used in conjunction with this video clip:
Don't kill yourself doing it. Should feel more like a casual workout. I've done it before for a bit, felt a little easy but saw great results in my 3rm.
A big part of the 5x5 program is deloads. If you do feel like you killed yourself, then you deload 10% next workout. If you fail to hit your reps two workouts in a row, you deload 10%. If you miss a week of training, you deload 10% (and 30% if you miss 2 weeks). You should never go to failure. If you did it counts as a failed attempt.
The progression is important too. You don't start 5x5 at a high weight. If you've never done it before, you start literally with just the bar. You won't feel challenged for a long time and it will feel easy, but like the guy said in the video, you will be able to keep it up, be consistent, less injuries, better form, etc... and the build-up will still give you some gains.
I used to avoid the deloads and ended up hurting my shoulder bad enough that I couldn't do shit for 6 months. Sprained my AC joint and my arm was in as sling forever. I actually thought I broke my collarbone.
Now if I don't feel I did well enough on a lift to progress and add weight next time, I won't, and sometimes I will even deload a bit. I don't care about the constant "only-up" attitude anymore. I know I CAN do it since I've done it before, so I can instead focus on being consistent. Maybe I'm not following the 5x5 rules to the letter but I am still enjoying myself and getting benefits out of it.