this is a VERY well known problem
my car shook like an earthquake once i hit 50, and it was gone at 60.
i changed my upper thrust arms with new ones, as is the thing to do when this happens, vibration totally gone.
if that doesnt completely fix the problem, you may need a wheel balancing.
only get your wheels balanced on a Hunter balancer with Roadforce, the E34 is rather picky.
and if that still doesnt fix it (very unlikely) you may have badly worn steering linkage, but as you said it feels secure so it probably is.
Alusil, Dinan DME | ITG Air Filter | Eibach/Bilstein HD | 26/20 Swaybars | Iridium plugs | Depo/ProLumen HID | Optima batt. | no AC | Stoptech brake lines, Frozen rotors, brass bushings, Superblue