I'm lolling at the link to the Irish waistcoats. Over a big belly? No!!
OP you're balking at the tucked in button down with jeans because it's a fashion mullet, way formal on top, way casual on bottom. You just need to adjust that level of formality with the fabric choice. If he's set on jeans at home, then a nice polo with a sport coat still looks polished for work from home. You could also do a crew neck cashmere sweater under a sport coat.
check out what the brits are doing:
https://www.instagram.com/efe.efeturi/