Well, of course, you are technically right.
But .....
IMHO (and I'm no expert in this domain), a road bike is something much more delicate than a mountain bike. And this could be a problem, if you have to go off-road even for a short distance, when bugging out. And you will certainly have some heavy BOB on your back too.

Nowadays, there is what we call VTC (Vélo Tous Chemins - "any road" bike), which is less sturdy than the VTT (Vélo Tous Terrains - your mountain bike) and faster, which my be a good compromise between the road bike and the mountain bike for a bug out bike.

I did ask me this same question (bug out vehicle) at the beginning of the year when strikes theatened to paralyse public transportation and automobile traffic in Paris. I felt very uncomfortable to depend only on my car.
So I recently bought a new "town" bike, just to put my a$$ on a bike again (which I have not done seriously for over 20 years ....). After a month, I enjoy riding it in Paris (late in the evening, when there is much less traffic).
And I guess I will buy a more robust VTC this winter or next spring.


Alain