I believe they are going to look at your delivery address and route it the best way to get it there - closer airport if that is the easiest. Might not always be.

Your bag not making it on the next GRB flight doesn't mean much - bags of pax traveling on that flight are going to have priority and their might have been no room. I had a bag delayed into CUN, day after Crowsstrike bug, and bag tracker was showing it was going to be on about 4 different flights to get it to CUN later same day. I watched my AirTag track the bag between the different gates those flights were leaving from, never to actually leave SFO (went between T3 and 2 about 3 times). Ended up eventually making the same flight next day, not any of the 4 previous flights (combo of non-stop and connections) it was apparently slated for.

Another time, we flew to SIN, and were flying on a separate ticket to BLR on SQ about 20 hours later. We gave the India delivery address since we knew it wouldn't make it to SIN in time...I'm not sure what routing it took (this was AC, that time), but it was delivered within about 12 hours of our arrival time. Clearly the system worked to route it onto a flight to efficiently deliver to BLR - I was surprised at how quickly it showed up given the circumstances.