Using a secondary climbing system is a great idea, been doing this myself for a few years now. I primarily climb DdRT and if I had to I am comfortable working with my line at odd angles, near horizontal, or even slightly below me at times depending on the situation, but will often choose to utilize a secondary system to improve work position.
As far as climbing without being tied in…generally not something I do. Aside from the potential risks if a branch you were holding onto or standing on breaks and you slipped, there’s also the psychological aspect of potentially thinking you’re tied in when you’re not and leaning back or letting go of the tree and not having any kind of support system to hold you. Although I will be honest and say that over the course of my career I’m sure there may have been a few rare instances where I was briefly not tied in while repositioning a lanyard or climb line. Definitely not saying that’s safe or okay to do either. I aim to always maintain at least one point of attachment. I did however do a lot of ‘free climbing’ (as in no gear or ropes) growing up. Climbing trees was one of my favorite things to do and I could easily spend hours up in the canopy reading or watching the neighborhood.