The car should have a key missing notice on the screen if the key is not nearby.
The Clarity, at least, does--it shows messages on the dash cluster that there's no remote present, and can only be dismissed for a few seconds. It doesn't ding nearly as frequently as I'd like, though. You would have to be quite distracted to not notice
eventually, and could just as well be ignoring other warnings (like out of fuel, door open, or check engine).
Shutting off in two minutes isn't
that bad, because basically you just can't get all that far away, although there are potential dangers so if I were programming that I'd give a countdown starting very quickly and with theoretically enough time to turn around and drive back to where you started before the car surreptitiously shut down.
Not shutting down at all (what the Clarity does) is potentially much worse if the driver is really not paying enough attention to notice the alerts, since you could theoretically drive for hours before parking who-knows-where, turning off the car, and realizing you have no means for turning it back on. Of course, that's no different from losing your keys in the same location, and also if you are ignoring no-key-fob alerts, you could just as well be ignoring no-oil-pressure or check-engine alerts that would have the same or worse effect.
Forcing itself into neutral almost immediately might be the least-inconvenient option (say, 30 seconds and a lot of insistent alerts after it loses the remote) so you just can't get far, but that one has the relatively abrupt side effect of what happens if the remote battery dies while you're driving in the middle of an 8-lane freeway.
There's basically no really good answer for the "drive away without the key fob" situation; one just has to accept the least-potentially-bad of a number of options in exchange for the ability to start the car without physically attaching the key to it. (I remember early Priuses and I think some Chryslers had a mode where you were forced to stick the fob into the dash to start the car, if you didn't want to risk it.)
Now, sitting in park with no remote is an
entirely different situation. In that case, the "never turn off" bad version is that the garage fills with CO and kills you when you go to get into your car. Which is pretty bad, and caused Chevrolet to issue a software update recall that shuts the car down after 90 minutes in park.
The "turn off automatically" version's bad is that you were relying on the climate control to keep the car from roasting your pets when you walked away, and it eventually shuts down and kills them. Of course, as implemented in the Clarity, since it is impossible to lock the car from the outside you're also trusting that no one will get in and steal your pets or other contents of the car, and since you
can shift it out of park and start driving (I just tested), you're also trusting that someone isn't going to steal your car, if this were intentional.
Honestly, what the Clarity does in park seems like a really bad design all around. If it indeed never shuts down, you have the CO danger (and running out of gas eventually); being able to shift out of park seems like a spectacularly bad idea with no upside, since your kids could put the car in gear or it could very easily be stolen; not being able to lock the doors was probably intended as an indicator or to prevent you from locking the key (or your pets) in the car or something, but in effect it just makes it that much easier to joyride, especially since the rapid-beep warning you get from the outside is the same as any other auto-lock-fail warning, which is pretty common, and isn't all that loud if you're on a noisy street so could realistically be missed pretty easily.
This is genuinely bad enough that I hope Honda changes the behavior with a software update eventually.