There are factors to consider and you could vary when exercising. These are intensity, volume and rest between each set. You are doing a low-intensity workout because you said that you could do them without difficulty. You are doing a lot of sets, seeing that you target to do 6-12 sets of 10 pushups each day but you're giving yourself too much rest in between that you won't be able achieve the area where you're pushing yourself. If you want to get stronger in the sense that you'll be able to do more push-ups per set, I advice you to change this routine and go for more reps each set or shorten the rest in between sets to maybe about two minutes.
My question is, would there be any benefit to "working out" like this?
There are benefits, yes. You'll burn a bit more calories just from doing the movement itself but the added benefit is that you raise your heart rate for a bit. This also trains the connection between muscle and your brain but a simple pushup is a simple movement unlike more compound movements.
Given that each set doesn't stress the muscles so much, would the compounded fatigue have any benefit?
Since you're not really pushing yourself even for the first six sets of 10 pushups with about 30-minute rests in-between, I'd say that you won't grow that much stronger. If you want to get stronger, you have to get out of the comfortable zone and strive to increase pushups for the last sets until you can't do them anymore. I'd advise you to do the last set of pushups for the day with increased reps up to the point where you'll just drop.