"GAMESS seems to have an old academic user base and a stable code for most traditional calculations"

GAMESS has been in development since the 1970s and split into GAMESS(US) and GAMESS(UK) in 1981, whereas MOLCAS has been in development since the 1980s and Psi4 is much newer. However, I would not focus to much on the age of the software. The first commits to the GitHub repository for PySCF were in 2014 but still it will be your first choice for certain things. The extra decades of history don't give GAMESS very much of a leg up overall against the other software packages mentioned.

"[GAMESS is good at] post-HF methods (and DFT methods),"

MOLCAS can do DFT as can Psi4. All of them can do post-HF calculations, with MOLCAS being more geared towards multi-reference methods such as CASSCF, RASSCF, GASSCF, LASSCF, CASOT2, RASPT2, etc., but still able to do coupled cluster and other things. MOLCAS also interfaces with NECI (to do FCIQMC) and QCMaquis, BLOCK, and CheMPS2 (to give 3 different options for DMRG), while Psi4 interfaces with CheMPS2 (to do DMRG) and MRCC (for arbitrary-order coupled cluster), so these programs can do a lot more of the more recently developed and closer to state-of-the-art post-SCF correlation methods.

"[GAMESS is good at] solvent continuum models etc."

MOLCAS can be used directly for solvent continuum models and Psi4 has an interface to PCMSolver for PCM (polarizable continuum model) calculations.

"Another good thing [about GAMESS] seems to be its ability to be interfaced MD packages like GROMACS with PLUMED (though I haven't used it)."

If you want something that can interface to GROMACS and PLUMED, then maybe this is the only place that you mentioned so far where GAMESS really has the other two beat, though the MOLCAS package has the DYNAMIX program which can do MD and it can bring you closer to quantum mechanical accuracy than GROMACS, via the surfacehop program and the interface to SHARC. As far as I know, Psi4 does not offer MD capabilities.

"OpenMOLCAS has a wide range of methods implemented but, I think its not as openly available as the other packages I have mentioned here. [...] Psi4 seems to be attractive as the source-code openly available to the community over Github and is actively being developed by a growing user-base."

The OpenMOLCAS source is openly available at GitLab here: I can't see why it should be considered any less "openly available" compared to the other two. GAMESS is apparently not considered "open source" and is just considered "source-available freeware" due to the way it's licensed.

"[OpenMOLCAS also has] a python module pyMolcas that can be used to create workflows. [... Psi4 also] has a user-friendly python module and can be easily integrated into any code/workflows."

While these things may be true, and you may consider them to indicate an advantage over GAMESS, I have yet to see any benefit to Python modules when doing serious quantum chemistry calculations, and I have done a lot of them. For GAMESS and MOLCAS, I would prepare an input file in VI and then run the software with scheduler via a submissions script. I can imagine a Python interface providing me with some benefits, and also some problems (in addition to those problems, I actually never managed to run a calculation in Psi4 because of its relationship to Python and the difficulties I had with installing it using the documentation's recommended process involving Anaconda and components which required Python versions higher than what I had available at the time).

"Many parts of the [Psi4] code seem to be written in C/C++ and features shared memory parallelization for multi-core machines."

Any quantum chemistry software worth discussing, has parallelization implemented for multi-core machines. GAMESS and MOLCAS are written in FORTRAN, which is specifically designed for scientific computing and number crunching, whereas C/C++ were designed for more general purposes, such as for operating systems, computer games, etc. FORTRAN is arguably much easier to learn, and large FORTRAN software packages like GAMESS and OpenMOLCAS tend to be easier to install because they don't have dependencies like BOOST and Eigen which are needed for C/C++ programs to do high-performance calculations (I mentioned that I've never been able to install Psi4 because of some Python issues, but I've also struggled with installing other C/C++ software due to the delicacies associated with BOOST and Eigen versions). See this: What is a good programming language for matter modeling?