Hmm, did you use the mod before? If so, check vehicle restrictions and most importantly custom lane connections. As an experiment you can load the savegame with the mod, go to options and at the Maintenance tab disable Lane connector and Lane arrows feature. If vehicles start spawning, it simply means there's a custom lane connection or lane arrow in the city that is breaking pathfinding for those vehicles.

In general the mod allows creating any lane connections you want, including ones that could effectively break pathfinding e.g.: by not allowing for certain turn at intersection, which in turn might completely isolate part of the city from the external world.
In vanilla, service building are not designed to work in "islands", because if one building cannot send vehicles to reach certain citizen to pick up the body, the game will try to send vehicles from other nearby buildings (game doesn't know if building is reachable before spawning actual vehicle).
If they also cannot reach the building, you will start seeing "flickering" of vehicle counter (0/1), which simply means there's a building in the city that cannot be reached by most or any of service buildings you have.
You can try Broken Nodes Detector mod to find such buildings, but it can be helpful if you experience broken node (vehicles disappear after updating roads, new roads are not used, vehicle speed suddenly drops when entering updated segment etc.).
Once you notice such behavior, simply run the scanner and delete/rebuild all pedestrian/bike paths it marked as broken (broken node of pedestrian/bike path affects pathfinding of all moving entities)

You were absolutely right! My road was inside a roundabout, but the connection of the lanes did not go there, and therefore no one could enter there. Stupid mistake. Everything is working now.