This is not true at all. My current company is privately held and has RSUs. Over the last two years there has been a single buyback and it was at a fixed closed market price and we were limited to selling 10% of our vested RSUs. Lots of startups flipped to RSUs with no plan to go public or be bought out in the last few years.
Additionally, even if your company is stable RSUs are frought with issues like what happened two orgs ago where a blackout period started and our stock went into a 30% free fall during the blackout and never recovered. We ended up getting a tender offer, the company sold to private equity and unvested shares were clawed back