I'm not Klonan (obviously) but I think I can at least suggest some things to make troubleshooting smoother (maybe we can even find the problem ourselves).
First disable all other mods and test if issue is still present. Vast majority of mods should be compatible (I have no problems with Krastorio 2 + AAI Vehicles + many other mods enabled) but maybe there is something that affects roads?
You should definitely check the network GUI (use shortcut button to the right of hotbar) and make sure requester and supply depots are in the same network (that is, both their road tiles are actually connected by the road). I haven't yet tried the second road tile so I don't know if they split the network, so I can only suggest to use the basic road (coal + bricks) at first, since I have no problems with that one.
Obviously, use the latest game and mod versions (I'm currently on 1.1.53 with Transport Drones 1.0.6).
If no issue with just Transport Drones (it should be okay to load even heavily modded save with just Drones enabled - factory might not work due ro recipe changes/power sources) but drones should be independent anyway (unless requesting new and not just changed item). Then you can slowly enable rest of the mods unless issue appears (you can do it in chunks, not necessarily one by one). Definitely look at mods that deal with tiles/roads, since Drones should otherwise work on mostly custom code.
Edit: I forgot about tiny issue I do have - road from Transport Drones is not compatible with Krastorio's Jackhammer (tool to quickly pickup floor tiles) but there are no problems removing/replacing it classic way (so hold any kind of paving tile in hand and place/replace/remove as usual) as well as using construction bots to do the work.