I was advised by one of the big smoke detector manufacturers that the real issue is voltage. Most battery-operated items will still run, more or less, on a much lower voltage than the nominal voltage on the battery label. That means that the battery can seriously run down and the gadget will still do something, even if it doesn't perform like with a fresh battery.
Smoke detectors aren't like that. They need the voltage of an almost new battery to operate properly. That's why they tell you to replace the battery at least once a year, even if a battery checker shows that it still has a lot of life. When you remove the "old" battery, that battery is still fine to use in other gadgets.
The battery doesn't sit idle in the detector if there isn't smoke. The detector constantly runs self checks, which use a little current. If you routinely test the alarm (as recommended), that also uses some current. So a year sitting in a smoke detector isn't like a year sitting on the shelf. The battery runs down a little and the voltage drops.
I've got to assume that the unit will still work if there's a fire and it hasn't produced a low battery warning. Fresh batteries typically don't produce a low battery warning in only a year, so there may be some PR compromise involved.
It's a pain in the butt to deal with a low battery alarm, especially if interconnected units all over the house decide to alert you in the middle of the night because one of the batteries dropped too low. So the one year time frame may be so that you can calmly replace the batteries at your convenience and avoid the "emergency" (and not hate the manufacturer). But what the unit considers a critically low voltage is still much higher than most other devices.
Which brings us to rechargeables. They typically have a fully charged voltage that is a little lower than the voltage of a fresh alkaline battery, or quickly drop below that level. The per-charge run time is also much shorter than the run time of a disposable alkaline battery. And as @ʎəʞo uɐɪ points out in a comment, rechargeables tend to have fast self-discharge rates and not hold their charge for a long time relative to smoke detector needs.
These characteristics aren't a problem in most gadgets. But it means that in a smoke detector, it may never be at the needed voltage even when fully charged, or will be above that level for a very short time. That's the main reason they aren't suitable.