Even if the US "guarantees" Taiwan security, the current administrations "guarantees" are not worth the even hot air they are made of.
I'm not sure if you were joking, but it would be 100% justified, especially if the "negotiation" with Russia over Ukraine are correctly reported.
South Korea and Japan are in basically identical places, FWIW. The only reason we haven't seen a north-east Asia nuclear arms race, so far, is that several countries relied on their presence inside the US Nuclear arms umbrella. With that looking more questionable, I think the next four years are likely to see widespread nuclear weapons proliferation. Certainly the example of Iran, where the US negotiated an excellent treaty and then unilaterally pulled out of it and started violating the treaty whole-sale under Trump's last time as president, and the fate of Ukraine, who were invaded by a much larger, nuclear armed neighbor, should encourage all three of those countries to make nuclear weapons faster.
The only reason any of them might not is domestic political response, which loom biggest for Japan, but I doubt that would stop any of those countries.
If I were them, I would try the Israel approach of very carefully refusing to admit whether they do or do not actually have 127 bombs that can reach Beijing.
Oh.. why 127? No specific reason… don’t worry about it…
If US lets Taiwan (or even Korea/Japan) into the nuclear club, then what is stopping nuclear proliferation to Iran? Venezuela? Cuba? There are negative externalities to US allowing it's allies to have nukes.
Taiwan doesn't want nukes according to the past Presidents.
ROC not having nukes is obviously also a policy choice by their leaders, not just a decision made in Washington DC, but if what is happening in Washington DC is different, I suspect that it will change the calculations in Taipei, Seoul, and Tokyo by a lot as well.
I think that a lot of baseline assumptions about how the world works are going to be proved invalid over the next four years, and I think the world will be worse off for it, but we're where we are now.
As for past President's statements, no one openly wants nuclear weapons until they have them. If you don't have any, then your public statements are that they are horrible, brutal devices of no value or use and you don't need them. Once you have them, they are vital to your national security. Citation: Every nation after the first one to get nuclear weapons(1).
1: Except South Africa, which never acknowledged them, and destroyed their weapons and the entire program rather than hand them over to Nelson Mandela. But that's a special case.