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.
+1
I'm in this camp on this one: commuted (did all my cycling, in fact) on 26" hardtails from 2002 to 2010, with suspension, switching tires as required. Still have my 2005 Giant Rainier (well, what's left of it -- the frame -- after upgrading things/wearing things out). With Marz. MX Pro fork, SRAM X9 d/t, BB7s, and XT/Mavic 717 wheels (all circa 2007-8) it weights around 26ish pounds. It's now been put back to mtb duties, but was a great 'dual-purpose' bike.
These bikes make very nice 'urban'/commute machines. While I agree with jyl that your 23 lb target is a bit unrealistic -- unless you were to throw a lot of money into the thing or strike really lucky on e.g. Craigslist -- 25/26 lbs all up isn't at all unrealistic. What I'd look for? Something like a C'dale F600/700, Spec. Stumpjumper, or Giant XTC from the early/mid 2000s. Just a few examples.
The only other point I'd make is re. tires. Not sure why the fixation on 1.25s; for a 559 ERTRO rim 1.5 or even 1.75 is a far better choice: lower pressure w/o risk of pinch flats will increase comfort and (if anything) will probably actually be 'faster' all else (tire quality etc.) being equal. I tried everything from 1 to 2" road tires, and settled on 1.5-1.75 as optimum (for 26" wheels). Used Panaracer Pasela TG in a 1.5 for years. Marathon Supremes are another great tire, as are these:
Compass Bicycles: 26" Tires