If one takes off with HDG SEL armed and no vertical mode (like VNAV) armed, at what height (AGL) will the vertical mode switch from TOGA to LVL CHG, and what will be the IAS applied? Also, why isn't it possible to arm LVL CHG before takeoff, as it is with VNAV? Depending on the entered plan, there are instances when VNAV cannot be armed, even if one wanted to.
LNAV & VNAV can be armed before takeoff; HDG SEL cannot be similarly armed.
If VNAV is armed & LNAV is not, then when VNAV engages at 400', the lateral mode switches to HDG SEL. But that's because the pilots had armed VNAV.
Taking off in TO/GA, the system won't revert on its own to LVL CHG. You can climb into the flight levels still in TOGA, although it's hard to think of a good reason to do do. If TOGA is active when the autopilot is engaged, the vertical mode will change at that point to LVL CHG, with V2 + 20 as the commanded airspeed.
As to why there is no option to arm LVL CHG on the ground, there is no need to. TOGA gives you speed guidance to bugged speeds (starting with V2 + 20) anyway, and VNAV gives you guidance to the profile speeds. Why offer another mode change that adds nothing to what you already have?