How to figure unusable sectors if the VOR is not colocated at/near an airport?
The US Chart Supplement will give you the necessary information you seek. The supplement lists a plethora of information on airports, airfields, heliports, NavAids, Flight Service Stations, ATC facilities, other aviation facilities, preferred routes, TEC routes, etc. You will find the useful and usability information for all of the NavAids listed in the entry for that particular NavAid. Regardless of if the same information is listed in the entry for an associated airport.
The US Cart supplement is generally organized in the following hierarchy:

Geographical region
State
City center
Facility name

I would imagine this is done to put the facilities in the same general geographical vicinity of each other grouped together in the supplement for ease of use when navigating in that area.
You can get the facility name from the either the VFR or the IFR sectional charts. Reviewing the information for the NavAid would be a good idea if you intend to use it for navigation. Similar to reviewing the information for the airfields of intended use.
As to why the unusable sectors of a NavAid are not depicted on the sectional chart? I would imagine it is just a matter of ease of use. The sectional chart is already cluttered with invaluable, important, critical information. Putting more information would get it to the point of overload and unusability as a navigation resource. Keeping it as an overview works well. Hence, the Chart Supplement is necessary to provide targeted information for your particular flight.

The unusable “sectors” of the VOR are not based on track, bearing, course or heading. They are based on the radials of the VOR. If you are on a radial depicted in the NavAid, airfield, or other facilities information, you must consider its usability.
If you are dialing/tuning into a specific radial, or if you are located on a specific radial (extended out to depict capturing the radial with a centered CDI regardless receiving thee signal) it does not matter where on the radial you are located.
Case in point. KLOL lies roughly on the 155 Radial of the LLC VOR. Yet, the airport lies on a 171° True Course from the LLC VOR with a 16° magnetic variation. The Wind Correction Angle will determine your Magnetic Heading. If the magnetic variation changes (which it does), but the VOR is not updated to the change, the radial will stay the same even though the Magnetic Course will be different.