There seems to be a fundamental lack of understanding of how the circuit works here. Please read the Wikipedia article: it's very good!
https://en.wikipedia.org/wiki/Multivibrator#Operation
Duh, you are right, I didn't realize the capacitors get their positively charged plate (since when does a cap have an "anode" & "cathode"?) connected to the ground when the transistor turns on, so they are effectively presenting a negative voltage to the base of the other one.
On the other hand, I have never seen this circuit implemented with the protection diodes - and it is a classic, possibly one of the first things all kids starting with electronics used to do (these days it would be probably an Arduino something but I am old school). This is the first time I see someone claim that the diodes are necessary.
How much is the breakdown a problem, really? When the breakdown occurs the BE junction behaves like a Zener diode (i.e. the breakdown is reversible) and the capacitor very rapidly discharges. The current spike could be large but for a very short time, so the thermal effects on the semiconductor (what could actually damage it) are going to be limited.
Here a guy actually measured a BC546 in the breakdown regime:
https://cr4.globalspec.com/thread/72501/Emitter-Base-Breakdown-Voltage-of-NPN-Used-as-ZenerHe used up to 100mA of current and doesn't report any adverse effects. There is another commenter there saying that most devices he tested couldn't handle more than about 3mA of reverse current - but that was also long time ago, with old components. Not sure how much does that apply to (semi-)modern transistors. And he also doesn't say for how long the current has been flowing.
I would say that as long as the device isn't getting hot due to excessive currents, this isn't likely to cause problems in this application.