It's not surprising that 35mm tires don't fit on a 20+ year old road bike. 28mm is probably pushing it.

If the bike fits (which is debatable considering the photos in the other thread) then just put some 25mm tires on and ride it. This idea that you have to have 35mm tires and flat bars/single speed to commute seems silly to me, but if you want those things - you'll need a different bike.