I don't know of prod-quality tools to migrate from jsdoc to TS. TS does have comment-level typing, so you can get the benefits without transpiling anything (similar to Flow).

TS pointedly does not minify output. It does the opposite: try to generate code that a human might have written.

It's very easy to incorporate TS into a Babel or Webpack build pipeline though, so you can use a purpose-built minifier of your choice.