Okay, to some level, they care; if you're down a lot, you have problems. But I'm talking about the occasional outage—something nobody thought was possible and had yet to pre-empt.
In those scenarios, your customers will accept it. Most will likely think, "Oh well, I'll just do something else". You've done this.
Everybody expects things to break: their car, their TV, their washing machine, their broadband. We deal with it - it's not that big of a deal.
Where this goes wrong is when you get ghosted. Telling your customers you're in pain, too, will reassure them. They want to ride the outage with you. When you have an outage and nobody is talking to you, you feel isolated; that is scary and frustrating.
Next time something breaks, bring your customers along. Engage with them and share that the outage is inconveniencing you, too.
Don't have an outage and ghost your customers because you decided to take lunch - or move house - or you're scared of admitting you have a problem in the first place.