Wood – How to square up slots cut with a circular saw

I don't have direct experience but generally speaking while a circular saw blade can go through nails, it's dangerous as it increases the risk of a kick back.

Although slower, you will be far safer if you cut the counter using a reciprocating saw with a demolition or dual wood/metal blade attached. That will cut through nails without the kickback potential.

You should cut your pieces a bit wide, say 1/8" extra, since your initial cut very likely won't be perfect. After the initial cut, you can sand the edge down to fit.

The traditional tool is a block plane. This is basically a chisel protruding through a smooth flat base that solves the evenness problem.

A variation on that is a tool like a Sureform. This is a hand tool that has numerous small chisel-like edges that shaves of curls of wood.

In either case, once ytou have removed most of the stock, you can finish the edge and get it smooth with the plywood by using a hand sanding block or an orbital power sander.

All of these approaches are a but slower and more work than a router, but do not require specialty tools (and give the special satisfaction that comes with working on wood by hand).