For me I start with mower brand, then engine brand. I stay away from briggs as much as possible. My experience with them is that they slowly lose there power under heavy use. Often parts go out faster on low end briggs causing the initial much cheaper investment into a much larger investment over time.
The bigger briggs engines are a lot better I don't see the power loss as much and parts are quite inexpensive.
I have used the older Suzuki engines which I absolutely love. The newer commercial kawi on the proline series, the big kawi on a scag both of which I liked except the smaller kawi seemed to bend crankshafts easy. I worked at the other place 5 summers and saw well over 50 cranks being replaced and only 6 go out every day.
My 160cc Honda is a beast won't bog down.
And I never used kohler so I am not sure.