So I have Some trains in LTN Depots with the leftover cargo (various ores). I've set up the Clean Up station but so far I've only managed to make the train visit it after forcibly sending it from unload LTN station while not empty. When I tell the train in depot to go to that depot (trying to trigger LTN basically), I get the message that train has entered depot with the leftover cargo. But that's it, it doesn't go to a clean up station.

Am I doing something wrong? Is there some other way to check if the trains in depot have leftover cargo that I'm missing?

So I have Some trains in LTN Depots with the leftover cargo (various ores). I've set up the Clean Up station but so far I've only managed to make the train visit it after forcibly sending it from unload LTN station while not empty. When I tell the train in depot to go to that depot (trying to trigger LTN basically), I get the message that train has entered depot with the leftover cargo. But that's it, it doesn't go to a clean up station.

Am I doing something wrong? Is there some other way to check if the trains in depot have leftover cargo that I'm missing?

You're not doing anything wrong. Cleanup warns if a train arrives at depot with remaining cargo, but doesn't send the train to cleanup stops.

This is due to there being only a few situations when it is safe to "remove" a train from the LTN network. One is when a train leaves a requester and the other when a delivery fails. If I were to send a train parked at a depot to a cleanup stop, there is a little chance LTN will use that train to make a delivery in the same tick (which can break both LTN's and Cleanup's dispatcher).

So to solve your problem.. You can make a "fake" provider and requester which will not load or unload any items and let LTN send all your trains through that stop. Since Cleanup tries its best to clean all trains leaving requesters, you should need to do this only once.

The other option is to disable LTN dispatcher (in mod settings), go through all your trains manually and send them through cleanup (by creating a new schedule). Again you should only need to do this once, after adding Cleanup to an existing save.