So far, it seems like my plan is working. There are still large sandy patches that hardly have anything in them; however, the clover has managed to spread fairly well everywhere else. It even made some inroads to the sandy patches, but our dry spring and summer left most of those clover pioneers dried out and crispy. Still, looking out into that part of the yard now, I can see most of it covered in trefoil.
My plan is to continue to let them flower, seed, and spread. I've noticed that the yard with the clover is that favorite for the hares and moose who visit our property. It also provides a comfy surface that does not require as much attention as a more traditional lawn. Considering the benefits that clover also provide to pollinators, I'm pretty happy with how it is turning out.