I do not blame you for wanting this to be better. Lord knows there are times I wish Illustrator was better at this. My answer is not meant to convey a sentiment that you are incorrect in your workflow or methods in any way, but merely explain why things aren't as you want them to be in terms of snapping.
Unfortunately paths of circles won't align to anything automatically themselves. Arcs are a difficult thing for Illustrator to align mid-path. Only the anchors along the circle's path offer any sort of real snapping. In other words, you won't get "a circle to intersect with other paths automatically". You can only get the anchors along a circle to snap or other paths to snap to the circle.
You can turn on Smart Guides (View > SmartGuides) and that will offer some snapping to the circle paths when drawing other paths. So, drawing a path you want as an intersect with the arc of a circle will snap the arc with Smart Guides enabled.
In addition to disabling Align to Pixel Grid, you may or may not want to enable/disable Snap to Point (View > Snap to Point) as well. If your circle arc happens to not be crossing an exact point, and Snap to Point in enabled, then your end anchor may not rest along the circle precisely. With Snap to Point disabled and Smart Guides on, you should be able to draw a path and have it snap to the arc of the circle.
The issue here...
.. is you are expecting Illustrator to align an anchor on the intersection of two arcs. Illustrator doesn't calculate that intersection. It calculates the arcs individually from their anchors, so it can snap to each arc, but illustrator doesn't see that intersection as a snap point. You have to manually ensure things align in such instances.
- Snap Paths to circle arc = yes
- Snap circle arcs to paths = no
- Snap path to multiple circle arcs = no
On the whole, Illustrator isn't always a precision tool like a CAD application would be. It's a drawing tool first, precision if kind of an afterthought in many areas. While it attempts to be precise, there are times when you need to use other features such as
View > Outline or
Object > Path > Average to get anchors to precisely line up. Depending on your goal, it is often necessary to take that extra step to align things. I don't fully understand the apparent issue with that. Precision takes a bit of extra work, as it does in any application.
If the goals is absolute precision, I'd suggest you work in Outline mode entirely for the set up. The strokes on paths can easily cause you the think something is aligned when it isn't whereas in Outline mode, you just see the actual paths.