I haven't flown the B737 series but the question applies to the Airbus type or any other design which incorporates versions of "LNAV-VNAV", (vice the usual descent methods).
The restriction on the use of LNAV/VNAV is usually other traffic and not any technical capability of using both to intercept the ILS, (or RNAV/GNSS) approach when so cleared. In complex terminals, the use of LNAV/VNAV to fly the SID or STAR reduces workload enormously but eternal vigilance is required to ensure routing, altitudes and speeds are flown as cleared. If an altitude or speed clearance is issued which is different than the SID/STAR, some form of intervention is obviously needed. For the Airbus series it would be Idle/Open Descent or Speed/Vertical Speed. For the B737 it would likely be FL CHG and a manual setting of the speed, (speed intervene, IIRC from the B767)
Because the bulk of traffic flies to common destinations, traffic/ATC requirements almost always require some form of intervention and so at some point in the STAR the autoflight will be used at one level below full LNAV/VNAV as described above. On the other hand, complex SIDs (Los Angeles, Frankfurt, London, Hong Kong, etc) are almost always flown using the aircraft's "LNAV/VNAV" capabilities.
You probably already have the
Boeing B737-8xx FCOM but of course this doesn't describe management of the operational environment.
PJ2