Crossfit has many issues:
1. It is not periodized and has no structured progression. The exercises are seemingly issued at random and have no natural progression. How can you really track your progress?
Depends on the gym, find the right one and you'll see them using hybrid westside and wendler's cycles.
Also progress is measured with the benchmark wods and 1RM if you really want numbers.
2. It is not specific. Crossfit issues general-purpose exercises designed to train you at random. While you may be fitter, it is not the ideal solution for someone looking to train for improvement in a specific area.
Fine, it is a general full body functional movement workout. Which in itself has its own advantages as well.
3. It is dangerous. A beginner cannot suddenly start doing olympic powerlifts and handstand pushups. A novice to intermediate Crossfitter probably can't do tabata olympic powerlifts (something that is in and of itself contradictory) until failure, then go for a 5 mile run the next day. The leg muscles will be fatigued and the legs will be more injury prome.
Every gym I know of typically has subs for any exercise that you can't do. So pike pushups from a plyo box instead of HSPU. Or scaling down weight on lifts to something manageable.
The fatigue thing is going to be up to you to listen to your own body. It is a group class after all with people attending at different rates.
4. IT IS DANGEROUS. On top of that, the typical mantra in Crossfit is to do as much as possible. The Crossfit mentality is that 20 bad squats in which you move a grand total of 4-5 inches is superior to 5 good squats where you go through the full range of motion with correct form. Repetitive use of poor form is a recipe for chronic injury.
Where do you hear this? I've never heard anything other than pushing form first. Several gyms I know where the coaches WILL cut your reps or otherwise force a sub if your form is crap in a metcon.
5. It is not for beginners. You can seriously hurt yourself doing some of the stuff Crossfit asks you to do, especially if you're not well-versed in the movements and lifts you're asked to do.
This is a repeat of 3 and same answer applies. Also there are typically fundamentals classes run for beginners to teach them the lifts. Additionally you will learn the progressions as part of workouts, and demonstrations are done.
Personally I'd say Crossfit is just a more intense version of p90x or Insanity where part of the benefit is getting out and having other people help motivate you through the workouts. As well as having access to some heavier equipment you might not be able to afford at home.
I definitely have seen improvement overall from doing it the last few months, and know others who are quite happy with their results as well, including absolute beginners.
That said, as with all workout programs... YMMV.
It is definitely mentally exhausting to push through some of the WODs. And as with all things, the coaching plays a large part of it. You can get terrible coaches, just like you can have terrible PTs, Massage Therapists or Chiropractors. All of which if they're good, you'll get benefit out of it -- if they're bad they could do some damage if you're not aware of your own body as well.
And for some the group vs solo workout can be the difference between actually pushing yourself hard enough to get gains.