So for something like this what would you even suggest as far as filling those gaps? Should I even bother?
Oh definitely replace the areas removed and fill the gaps. As you linked earlier you can buy 3.5” fiberglass batts and gently push them into place (wearing gloves, long sleeves and a mask of course). That will put your walls into about same state as before.
SLightly better for your situation would be mineral wool batts (often called by the manufacturing name ‘Rockwool’). Mineral wool can be swapped out for fiberglass, is slightly denser, completely impervious to flames and water, and performs better when there’s lots of air movement. It’s a bit more expensive, but given the relatively small amount you’d need (one bag) it’s what I would do.’
As for improving the home’s insulation overall, that’s another rabbit-hole. Finding out where and how air is moving through is a big first step, and a blower-door test would help and is the gold standard to know how ‘leaky’ your home is. If you have siding that’s relatively easy to remove and replace (like vinyl siding) you could take it off and replace it with a much better house-wrap (there are dozens, and which one to use is another whole point of discussion - but almost any would be a marked improvement). But if you have something like stucco ont he exterior (which I suspect given the location and the clay tile roofing) that isn’t an option, and you’d probably have to address it from the interior, and it probably wouldn’t be worth it economically.
It is still a very good idea to chase down any and all sources of air-leakage, and seal them up. While it takes some time and effort, the materials are incredibly cheap - Caulk and Great-Stuff (foam in a can) are your best tools. Check every penetration (windows, vents, cable-routing) and ensure they are sealed (most probably aren’t).