I've never seen a briggs with the primer bulb built to the carb before except on their old plastic carbs, i know Tecumseh used that style/design quite a bit.
anyhow, that engine appears to be a L-head a.k.a flat head, so it looks like that was the valve cover the took off. if he takes it off again he should see the springs and valve stems in their. i would replace that gasket too.
EDIT: According to Toro that mower uses a Tecumseh LV195EA
on the throttle linkages, this video might can help you out,