Any way to fill these shapes with a pattern somehow, and avoid that duplicated line system which looks like a total mess in the keyline view below? Any cleaner way to fill shapes with a pattern like this?
Ideally looking for a solution that works with both rectangular and circular shapes as seen below.
Use a pattern...
There are a bunch of line patterns loaded with Illustrator by default (Open Swatch Library → Patterns → Basic Graphics → Basic Graphics Lines).
You can use them as a second fill using the appearance panel and use blending etc to get the effect you want. You can add a Transform effect to that specific fill (make sure to check "Transform Patterns") to get the rotation & scale you want:
...and the same with a different blending mode:
If the default line patterns don't work for you then you can of course make your own pattern; which should be as easy as creating a small section of the lines you want (you could do it with a single line if you really wanted to) and dragging them to the Swatches panel, then double clicking to enter the pattern editor:
Read more here:
You can create a Pattern Swatch and use that pattern as a fill. I would start with what you have already, with two specific notes:
Then create a clipping mask the same size as your background
At this point you can drag the whole clipping group into the Swatches Panel.
When you assign this swatch as a fill to your desired object, double-click on the swatch to edit the options. Select the "Size Tile to Art" option and you're done.
