Can a cordless jigsaw replace a corded one for woodworking?

For most home and trim work, yes. Reviewers at protoolreviews.com note that modern 18V/20V brushless jigsaws like the Milwaukee M18 FUEL and Makita LXT cut cleanly through 8/4 hardwood. The main tradeoffs the reviewers we read flag are weight (cordless tools with a 5.0Ah battery get heavy overhead) and the upfront cost if you're not already on a battery platform.