They do care, %100 they care. If you take longer to do task X because the SaaS solution crashes or is unavailable, or causes issues in finance, or a dozen other things then the company will very much care. I literally work at a SaaS company and hear complaints from clients. Money is all that matters, if your solution isn't as good at making/saving them money as another solution, you get dropped. And reliability is a big part of that. A solution that frequently has issues is not a money-making/saving system that can be relied on.
It's not about looking like a fool; it's about what your P&L looks like. That's what actually matters. Say you made a nice slide deck about product X and got buy-in. Walking that back is MUCH easier to do than having to justify a hit to your P&L.
What experience do you have to be making these claims?