I wish I had all the answers to training. If anyone is watching puppy peaks you could see how carefully SG building tug and other toys drive in her 4-5 weeks old puppy so yes, it is a learned behavior and not something puppy is born with. Sure some dogs have higher drive and will naturally retrieve, tug and play with toys and some need more motivation but motivation it is. Is it a big problem if your dog only food motivated?- it is all relative but it is much easier to work with dog that is both toy and food motivated. Because food causes satiation which known as cookie coma
If you have low tugging element, try for next couple weeks carry small tug with you and start playing with it at the moments of highest arousal for the dog- i.e. if you dog goes nuts when you return home, or at the sign of leash because its means going for walks or when dog is about to have zoomies etc
Pair up your toy or tug with highest level of arousal that dog has and see if that helps.