Having worked on and installed autopilots in the past, you are generally correct about GA AP systems. They’re not like airliners or jets designed to be engaged very shortly after takeoff. In the GA world the systems are not typically sophisticated enough to null out large control surface deflections and you’ll sometimes have systems attempting to max out elevator trim and disconnecting the whole AP.
I have yet to try this out in the G36 to see if it works the way I would expect it to, but then again I’ve never worked or flown on anything with avionics like the G1000.
It is also good pilotage to only engage the AP once the aircraft is in its stabilized climb (trim included). I tend to only ask autopilot to hold what I’ve done, rather than genuinely fly the airplane for me; I’ll do the majority of my pitch changes by hand, then reengage the system so I can focus on other tasks. In the real world, autopilots are often finicky and break kinda frequently, so you learn to use it as a pilot aid, rather than depending on it to fly for you. Like cruise control in your car, it’s nice to have, but you shouldn’t need it to drive on the highway.
I’ll also add how I use the FD/AP in the 172:
On the ramp, I’ll set my altitude, runway heading bug, enable the FD, and set the FLC for the enroute climb airspeed. The command bars activate and frig off to somewhere. I’ll line up, takeoff (using Vx or Vy as appropriate) and get to at least 500’AGL before I start looking for the command bars again. Once I’m at about pattern altitude, I’ll nose down to get my enroute climb airspeed (matched up with command bars by this point) and engage the AP. Then I’ll move the heading bug to fly me to the route, only engaging the NAV once it’s almost perfectly lined up.