No, I am not flying a 737. But, the FMS I use gets its position information from radio aids and if no suitable radio aids are available from GPS. From looking at the nav display and the results e.g. being very close to the track, it does a fine job. Still better than trying to fly the needle which swings back and forth all the time. And, coupled with VNAV it flies just like an ILS.
Also, I would imagine that you have to keep your fingers on HdgSel all the itme, since the wind changes with altitude. Anyhow, the workload is reduced and more time is spend monitoring.
Isnīt there an option for GPS on the 737? Now, I am not suggesting GPS as a sole means of navigation, but in most cases it is a lot better than beacons.