A couple of J-15s have crashed unfortunately and the initial instances were rumoured to be due to FCS issues in very rare flight parameters. Some have also been due to bird strikes and the difficulty of practicing carrier landings/ takeoff (I think there was one). None of these were hidden from the Chinese public or the world mostly because there's nothing much to be embarrassed about. These things are insanely difficult like you said and practice makes perfect but PLANAF have not got the experience on more mature carrier operating navies. I think even Russia lost a few Mig-29k and Su-33 in training and operation as well as the USN in its own early days (with yesteryear's planes) before they got the carrier ops and procedures right. Learning curve is steep here for PLANAF as well but one must start somewhere.
However all that said, J-15's FCS issue that has caused at least one of these crashes may or may not have been resolved yet. The plane itself is awesome on paper. A CATOBAR ready J-15 is going to be akin to an F-14 with modern avionics. Payload, range, and flight performance are all going to be top shelf even if it runs into the huge problem of 5th gens in the pacific. I think those qualities will still have a place in the future despite 5th gen proliferation and further. Perhaps new developments can balance the playing field a bit and certainly made 4th gen fighters very much relevant into mid century. So PLAN will not abandon this beautiful resurrected Sino-Su-33 until clear superiors are cheaply and quickly available for PLANAF. This sort of forces SAC to address the FCS problem or lose PLANAF orders and at the moment, if not J-15, SAC don't really have concrete orders that we know of yet. I'll believe J-31 when I see solid confirmations.
All this means SAC will have been pouring a lot of energy into resolving the FCS and I'm sure they would have gotten it or reduced it to such a degree that PLANAF pilots are now intimately aware of how to use the J-15 if the problem persists. Was all probably a huge FCS oversight and one of those instances where only meeting the problem allows for awareness and then resolution while the solution is very straightforward and simple. Unfortunately there are many cases of this in the aerospace industry throughout the world as much as engineers put tremendous efforts in thinking of everything. Saving time and resources by copying a Su-27K prototype from Ukraine had consequences for PLANAF in this case. RIP and respect for the pilots that PLAN chose to take some shortcuts. Then again all of this is very new to SAC and PLAN. Hopefully a navalised J-31 is purpose built and draws from all the lessons learned from J-15. So far always rumours of PLAN order.
Though AFAIK, we do NOT have a solid reason to believe that these rumours are 100% accurate and FCS is, by any means, flawed or unstable. Both Liaoning's and Kuznetsov's CAGs suffered from similar incidents under similar circumstances (at least in some cases) which might also suggest that the problem might be more related with the ship class than we acknowledge.
Back in 2016, Kuznetsov suffered twice -- one MiG-29KUBR and one Su-33, just in three weeks. Given the practical differences between both navies when it comes to CAG Ops, one could propose that it's natural to expect more incidents from PLANAF since PLAN sees carrier operations more crucial to achieve their mid-term goals and Russian Navy had to partially abandon and fully postpone those plans until 2021+.
Both navies share 3 factors in common:
1) Related aircrafts in their roots; Su-33 & J-15
2) Sister ships in class with very similar characteristics
3) Lack of experience in CAG operations as of 2016
and in my opinion, the 3rd factor is quite a strong one when it comes to such incidents. I'm in no place of eliminating the possible reasons of such incidents but I'd pick experience over the others any day. That being said, PLANAF is climbing the ladder much faster than anticipated by the most and this is also related with an accelerated but intensive training program. Such an approach often create incidents related with human factor.