It sounds like you have the Global Map selected. If so, it is not routeable hence the straight line. If you are new to BaseCamp you need to install a routeable map or connect a device and use its map. If it has worked in the past then go to the Maps pulldown menu and select a routeable map.
The only other thing I know of that creates the straight lines is when the starting route point of a segment is assigned the Direct activity profile. This doesn't apply to your case though since a "Direct" route point would behave the same in all Activity profiles. I am at a loss to explain what you are experiencing. At a minimum I would compare the routing settings in a working Activity profiled with one of the non-working profiles, although I don't see anything in there that would explain your direct routing issue.
Not lockable just means it can be used anywhere, but as it's an OSM map that's probably the issue. Try contacting the map maker to find why it's not routeable in BaseCamp when the car activity is selected, or simply use another activity that does route correctly.
Third party maps do not always designate features in the same way as Garmin. This can cause routing to fail and default to Direct Routing (straight line point to point). Some maps will target off-road activities like Hiking - trails are routable, but roads may not be routable.
If the routes you observe with the Hiking or Walking profile are creating acceptable routes, then there is most likely a feature type avoidance that is preventing the Driving profile from working.
Try creating a new Route type by going to Edit>Options, select Activity Profile, set the profile to Driving in the drop-down list. Click on the + symbol to the right. Give it name (like Driving2) and have it based on the Driving profile. With the new profile go to the Routing tab and deselect Feature Type Avoidances. My guess would be Narrow Paths and/or U-turns would be the issue.
Different Profiles use different settings to best suit a particular activity. But a profile works for your purpose, it doesn't have to match the Activity of the profile. You will need to match the Profile of the Route on your Device. If Driving gives the desired routing for a hike, then it is okay to use the Driving profile.
