Value depends on what you grow and what those plants need most. Raising the ph will increase your P and K availability. :banana: Even grass likes to be above 6.
All limes are not equal. There are other factors In addition to the ph effect of grind size . Some limes are higher in calcium, sulfur... Industrial by product limes often have the highest value, but can clump up and be difficult to spread evenly.
Working it in is the standard advise. For sod I would apply a ton of fine grind per acre per year till I got to the target ph. The lime will slowly work down in the soil.