Why Taiwan modernized Gearings after the US stopped

Was the proposed further FRAM modernization of Gearing-class destroyers essentially what Taiwan later attempted?

Broadly yes, but Taiwan's less-worn ships and smaller fleet made a deep rebuild more worthwhile than it was for the United States.

The American proposal ran into machinery and hull-life limits after decades of harder US service. Taiwanese ships had not accumulated the same wear and represented a much larger share of a smaller fleet's combat value. For the United States, applying an expensive conversion across many old destroyers could lose out to buying new ships. For Taiwan, rebuilding a smaller number of valuable hulls could be easier to fund and strategically worthwhile even at a similar unit cost.