The cooler is one of the more popular mash tuns for homebrewers. It's cheap, and extremely easy to assemble on your own. I built mine for under 100 USD and I get around 80% efficiency and it maintains the mash temp. extremely well. Also, what you described is exactly what a false bottom is. I use a bazooka screen, but I started with a mesh hose that you can get from a local hardware store.
Congrats on your first AG adventure!
Primary 1: IIPA(Simcoe/Citra)
Primary 2: California Common
Secondary 1: Infected batch with Lactobacillus
Bottled: HBC 342 XPA, WLP670 Saison
Kegged: Orange Blossom Honey Ale, Red Ale
I shotgun heady topper.