No, it's much more direct than that.
Specifically, this sentence, while technically true, is highly misleading:
This is true (in the short term, see below), but benzodiazepines increase the action of GABA, causing chloride channels to open in the presence of GABA to a greater degree than they do otherwise. So, without actually modulating the amount of GABA in the brain, the GABA which is present has the same effect as a greater amount of GABA would in absence of the benzo... "virtual GABA", so to speak.
In the long term, the brain becomes acclimated to the increased action of GABA, and responds by downregulating GABA receptors and potentially even shorting the supply. This is why I think benzos are, eventually, a worse-than-zero-sum game: if your problem (tinnitus, seizures, whatever) is caused by, basically, a GABA insufficiency, taking benzos for an extended period will eventually make your problem worse.