Does a dead battery on the FOB key prevent the car from starting due to anti theft system?

I suspect that it could but you should see :"security" blinking on the instrument panel. Did you pry the plastic off and look for the battery? FYI Home Depot has most car keys in stock and the charge about $60 for the key and programming

Systems do vary from car make to make and from system to system, but it certainly could.

The battery is easy to change. Take the fob apart, pop out the lithium ion that is in there, and replace with a new one. I get mine at dollar stores. If the battery is dead, you can still start the car by holding the key fob right next to the car's start/stop button when you push it.

https://youtu.be/bMdL502jAsU The Smart FOB key design is identical for various Honda models including Odyssey / Pilot / CRV/ HRV / Accord / Civic / Ridgeline. The steps shown in this video can be used to replace the battery on the models 2013 - present depending on the model.