Schools are absolutely fine until they're not though. In my current school, no-one is taught science by someone without a science degree, although it's not always in the subject they are teaching. However, when they last had to recruit, they had to advertise multiple times, and they could easily have ended up with an unfilled vacancy. They got lucky with someone looking to move schools at the last minute in the summer.
It's a nice school, in a cheapish town with good transport links. The latter two are relevant because it means an ECT can afford to live here, and people commute in from quite a large area because it's relatively easy (decent A road, mainline train station etc). We're still finding it harder and harder to recruit. We have good links with ITT programs at the moment, so it's easier to recruit ECTs as well.
I've equally worked in schools in more expensive areas where it's hard to find a rental property, and you can't get to the school easily without car. That quickly limits your recruitment pool.
I've also worked in a school where the science department was fully staffed with specialists when I joined. And it was fine for a couple of years. Then our HoD left (for a promotion) and the person they recruited to replace him was a lot less experienced and it caused some issues. At the end of that year, we lost a full time member of staff, a part time member of staff and someone went on maternity in about October. We didn't manage to recruit for September, and it was chaos. The pressure put on those of us who were left was completely unrealistic.
Amazingly, we did manage to recruit mid year, but because things were so bad, people were leaving whenever they could. So we lost 3 members of staff, that year and only recruited one. I couldn't face another year like that and left, another friend left at Christmas the following year.
In departments like that it can become a vicious cycle- we're a few years down the line and they're still short staffed now. Most of the staff I worked with in that department have left. I think the department is in a better state than it was, but the school is still advertising for science teachers. After a while, when you see the same adverts come up again and again, teachers start to assume there is something wrong with the school. But 5 years ago, it was fine.
I don't think any school is immune, especially if something happens like a change in leadership or a bad Ofsted.