Is it possible to record Quick Time video of iPhone that is rotating?
The reason this happens is because a video cannot change orientation mid-record.

IE: My wife starts many of her video recordings on her phone in a
vertical position. I remind her that we will likely have a terrible
experience watching that video at a future date from a television
because of the huge black bars on the left and right. She rotates it
mid recording session, but all this does is make it so we have to turn
our head on the couch.

There's a reason you haven't seen anything rotate on Youtube either -- it simply isn't equiped to all the sudden change from widescreen to portrait.
Solution:
Quicktime (assuming you're on a mac based on your screenshot) also lets you select &quot;New Screen Recording&quot; from the File dropdown menu. You could record your whole screen like you alluded to, but if you want a more controled/correct size, you would do the following:

Pretend like you're going to do a new iPhone recording. Have the phone on the screen. Take note of where the phone is on the screen in portrait mode, as well as landscape mode.
Use sticky notes on your monitor to create a &quot;square&quot; that will make sure your phone is completely visible during landscape as well as portrait. (see where I'm going with this?)
Tricky part. Open your applications folder. COPY quicktime to your desktop. Double click the desktop icon. This will open a SECOND instance of quicktime.
Use the second instance of quicktime to do a &quot;New Screen Recording&quot;. You do NOT have to record your whole screen; Select the square that you made with sticky notes on your monitor so we record this area.
Hit record on the second instance of quicktime, while the first instance of quicktime is simply used to watch the phone in both portrait and landscape!

I know, it's a little involved. Likely easier to buy some software, but for free -- you can't beat it.