Yes and no. The key here is are the trash cans regularly emptied by the park employees? If yes - they can be a great asset. If no....they are inviting disaster. Thinking of 2 courses I've played with lots of cans. One is in a state park with cans built into every tee sign and these are emptied daily in the summer by the parks dept. This is one of the cleanest courses I've played. On the other end...another course I've played has cans everywhere and they are only emptied by well meaning DGers. In reality, they are usually overflowing and folks take that as an invitation to just chuck their crap on the ground next to it. It is always a mess.
Yeah, this is going to vary regionally. The OP will have to consider the likely demographics of this park/course and make an educated guess which will work better. They could try the "carry in/carry out" no cans tactic first and see how littered up it gets. IME there is going to be litter regardless, it's just a matter of whether it's all over the place or somewhat around the vicinity of trash cans. :\