They don't want to control reactivity anymore, they likely want to kill the reactor core completely in order to not take any risks (not just to keep the core subcritical, but to avoid it simply melting as well, if the pumps did not work they don't have the waterflow they need, so the next option is to end the core fast). And for that, boron is the best tool.
Subcritical is "killed completely" from a nuclear standpoint - lowering the reactivity of an already subcritical reactor (e.g. adding neutron absorbers) does nothing but make the fission rate decrease faster. The fission rate already changes on short time scales, (e.g. orders of magnitude in minutes/hours during a scram). If the fission rate is already essentially zero, which it is in the Japanese reactor by now, it does nothing at all.
The heat inside the core is being produced by the decay of fission products, and there's absolutely nothing you can do to stop that except wait for enough half-lives that the activity slows down a bit.
Your general thesis, namely that as soon as they decide the reactor can't be salvaged, they'll dump in a bunch of boron just to be safe on the nuclear front, is correct, but where you're wrong is in assuming that adding a bunch of boron will help cool the reactor at all - it won't. All it will do is ensure that the reactor never goes critical again.
This is why spent fuel needs to sit in a pool of water for 5 years before anyone even considers moving it. Decay heat is serious shit, and it's not related to neutron physics at all.
EDIT: Probably time to get more specific about the term "reactivity". Reactivity is related to the "multiplication factor", which tells you how much bigger each generation of neutrons is than the last. It's zero when each subsequent generation is the same size - this is the normal operating state of a reactor, positive when each subsequent generation of neutrons is larger, and negative ... you get the idea.
If you've ever touched population dynamics in a differential equations class, you'll realize that this is a recipe for exponential growth and decay. Basically the time scale on which nuclear reactions proceed is "reactivity / mean neutron lifetime", with the caveat that if your reactivity is just above zero, neutron population growth is constrained by the longest-lived neutrons (it's like if every family has 2 kids, except for a couple hundredths of a percent, who have three, but put their third kids in cryostasis for a thousand years). In practice, this is how reactors are operated, because the mean neutron lifetime is _very_ small, so the worst thing that can happen is if your reactivity moves outside of this regime (this happened at Chernobyl). If your reactivity is negative, you are _always_ constrained by the longest-lived neutrons, but that's okay, because even this time scale is pretty short.
The upshot here is that if each successive generation of neutrons is smaller than the last, the number of neutrons (and hence things like fission rate that depend linearly on the number of neutrons) decays exponentially with a pretty short time constant.
The definition of "subcritical" is "having negative reactivity", so if the reactor is subcritical for any length of time, the fission rate will have exponentially decayed down to a tiny number.
