Best way to kill off lawn?

For spring I have a lot of plans around the garden and one of those plans includes destroying existing lawn (or weeds tbh). I want the entire lawn killed off so I can till the soil, introduce some new soil/sand mix and start again fresh with seed in the spring.

The lawn is just absolutely too far gone with Paspalum and other disgusting weeds. I had heard the following suggestions already.

1) Glysophate the lawn and it’ll all be dead 100% dead within a month.

2) Cover the lawn in cardboard and leave for 1-2 months

I prefer the glysophate option however I’m worried that the concrete edging its about 20-30cm deep won’t be enough to keep the glysophate from penetrating deep into the soil and the feeding runner roots from the plants I want to KEEP might be deep under the concrete edging and suck up the glysophate and die.

Thoughts?