What you describe would be exactly reason #1 I would immediately say to anybody on topic of why not saving desperately on such a thing, despite never being in such situation myself.
Sometimes learning from other's mistakes should be enough. No, mostly it should be enough.
The immobilizer is the single best piece of technology for preventing car theft. If you create a backdoor for bypassing it, you'll end up like Hyundai/Kia which decided to sell cars without the immobilizer in recent years and which have turned into a joke in the minds of potential customers.
It does not require a battery in most cases and is separate from the keeloq system that controls your car's doors.
When that battery dies, you can press the directly to the start button and it uses a “receiver powered transmitter” RFID close proximity to start and run the vehicle.
Most people don’t know this, so when that battery dies they panic and suffer.
In my experience virtually everything made in last 15 years will either support this RFID backup or have a spare physical key hidden inside the keyless fob.
Lots of them will even let you press the dead key against some part of the exterior to unlock the doors too.
They just left that same antenna in place (in Chrysler+ anyhow, called SKIM) now as a backup instead of the primary.
Edit: RFID should not need battery…you can reprogram your own key by jumpering the correct OBD2 pins..process takes about 20 minutes…
you instead need to hold the fob up to the start button and it will work passively, rather than just being in the car normally. Glad they still give manuals with cars as I had to learn that without service.