What I don't get is why we don't just go ahead and recognize them.

I've read something about that. They perceive a benefit to ambiguity.

Strategic ambiguity was good for a time, but things may have changed enough to where recognition and an open declaration of support may be warranted. It was/is a balancing act.

SNIP Strategic ambiguity
on Taiwan is the long-standing U.S. policy of intentionally being unclear about whether it would militarily intervene if China attacked Taiwan, aiming to deter both Chinese invasion and Taiwanese formal independence by keeping Beijing guessing about a U.S. response, while also signaling support through arms sales under the Taiwan Relations Act. This policy creates uncertainty for China to discourage aggression and for Taiwan to prevent provocations, but its effectiveness is increasingly debated as China's military power grows, leading some to call for strategic clarity (an explicit defense commitment).
Key aspects of strategic ambiguity:

Deterrence: Prevents China from invading by leaving open the possibility of U.S. military involvement.
Restraint: Discourages Taiwan from declaring formal independence, which China views as a red line, by withholding a guarantee of U.S.
military rescue.
Uncertainty: Deliberately creates ambiguity about the conditions and nature of U.S. intervention.
Legal Basis: Stemmed from the U.S. recognizing the PRC and ending formal ties, but the Taiwan Relations Act (TRA) established unofficial
relations and mandated defensive arms sales, creating this ambiguous stance. SNIP