Hello Muzialis - Good to hear from you again!
I used to care about polyconvexity back in my PhD student days. Since then I have developed a more practical approach when it comes to FE modeling. My experience is that mMost reasonably formulated material models have little issues with stability. There are certainly cases when instabilities can occur, these are often effectively address using numerical damping or contact stability. In other words, I have not seen real problems based on lack of polyconvexity.