I commuted for years on a vintage hardtail mountain bike (old Bridgestone) with 1.5" road 26" tires and a vintage suspension fork (old Rock Shox). It was a great commute bike. The sprung fork wasn't necessary but wasn't a detriment either.

Something like that is pretty common on Craigslist, if you are flexible about the brand. I also see older Cannondale mountain bikes with Headshock suspension pretty often.

1.25" tires and 23 lb may be a stretch but the general idea you have is good.