I'm not really sure what you are asking, OP.
On the one hand you are asking about limited involvement for a simple problem like holding a pencil incorrectly. Of course any sane and rational person would say grab some weekly therapy, and why would you need home visits, IEPs, doubled-down on private/county services. To even ask is ludicrous. So presenting the problem on that level is kind of silly.
If however, in fact, you have a child with actual delays (which is what we all assumed you meant) then I think the solid consensus is that most people consider that county services offer a good foundation (brought to your home, which is super convenient), and that supplementing with private therapy enriches and doubles your therapeutic gains. Very young children often are successful at this stage from the double dose of private/county and overcome their delay so even thinking about Kindergarten/IEPs/permanent records is just spinning up things in your mind.
Try to stay in the present and with the problem. It sounds like you have a child with delays, and a young child. He could benefit from therapy, and the county offers therapy free at your home. Supplementing with private therapy offers additional benefits. Why not do both if you are lucky enough to have good insurance. Worrying about paperwork and IEP meetings sounds like your anxiety talking, not what's good for your son.