3-speeds are serious bikes and are anything but obsolete. They are bomb-proof. I just bought a '64 Raliegh clone (they are all mostly Ralieghs, basically). Found it for $50 in a bike shop. Very dirty. Took it home, cleaned it up and pulled the hub apart to clean that. After rubbing compound, wax, WD-40 and fine steel wool to remove the minor rust specks on the chrome, it looks like new. Re-packed the bearings but left the BB alone since these all had cottered cranks. Now the problem is that it doesn't look like a beater anymore, but rather, a restored beauty.
I use it for under 10 miles round trip quite often. I have a Brooks B73 on order for it.
If you go for one of these I will pass on a word of caution: Do not stand on it in 3rd gear. Sooner or later, even a well adjusted AW hub will slip into neutral due to a design flaw. If you are bearing down while standing in 3rd gear you WILL face-plant over the bars! They are not designed for racing but for commuting and short trips although they are not limited to such.
You will find that these things go pretty fast in 3rd as they are geared high. My area is hilly but it takes them ok in 1st gear.
BTW, that Trek 950 or 930 you found is no beater. I have a Trek 830 (1988) that continues to give good service and is morphing into a better bike. It's what I use as my regular.
Up the British!!