I use a 5 gallon cooler and mash 5 gallon batches in it but there are limitations.
I just did a brew last night and with 11.5 lbs of grains and 4 gallons of strike water is was pretty full.
I do a lot of experimental 3-4 gallon batches and it works fine for that. 5 gallon batches work OK until you want to go to higher gravity brews.
If I reduce the water/grain ratio to 1 qt/lb, I can mash about 14.5 lbs and not spill when I stir it. That's not an ideal situation, but it can be done. The 5 gallon cooler is less money, and being and maybe is a little easier to rinse out using the kitchen sink, but the 10 gallon cooler is really the better choice as it gives you more options.
I just added a spigot, a stainless steel 90 degree elbow pointed down and a BIAB bag and it works fine.