This is VERY true !
I have been boxing for 5 years and never did I get injured. I have been training wrestling and bjj for 3 years and I already sustained 2 concussions and a torn mcl.
While competing in striking sports is very dangerous, training in striking is very safe because you have control on everything that is happening.
If you do touch or light sparring and don't throw punches hard enough to concuss one another, there is literally nothing that can happen except maybe minor facial injuries.
Now, when it comes to grappling, especially standup grappling, you haven't much control on what is happening (except if one guy is way better than the other and is careful…) thus neck/spinal injuries, and even concussions, are likely to happen.
Moreover, we spar way more in grappling sports than we do in striking sports. In addition to that, there is this common misbelief that grappling is safe, so trainers aren't so aware of the risks, while they are very aware of them in striking sports.
And this is from an "accidental injuries" perspective, now if you are talking about overuse injuries, they are obviously way more common and worse in grappling sports than in striking sports.
So, from my own experience, striking can be very very dangerous but also very safe. The safety gap between competing and light training is huge.
Grappling is Always kinda Dangerous (especially wrestling and judo) and the gap between competition and training isn't very big.