Last week (doesn't help this week) had a 460 piece for running 4 mythic 0 dungeons. The Heroic Darkshore Warfront guarentees a 460 piece as well, and this rotates I think once every month or less?

Visions have been a great source of ilvl gain as well, but working them up takes a bit as you won't get good rewards until you can fully clear.
It's important to try and run mythic + / visions because you get currency from them. The Titan Residuum has 450 / 465 / 480 pieces that are all quite affordable as time goes on, it's the main way I've upgraded my azerite gear (head/shoulders/chest), as the 450 piece is I think only 175. So make sure your scrapping the pieces (head/shoulder/chest) you don't use. Once you complete a higher mythic+ though it gives you a ton more.

The Emissaries will start rewarding 445 gear once your ilvl is a little higher up.
Can always farm gold and see if there is any pieces that are cheap in the AH.

You'll eventually hit a wall though and the only option is Raid / Mythic + / Visions. As right now most of my teams are in the 465-474 range. Things get further complicated because you then need the right secondary stats on items and the right corruptions, so it feels bad when a 450 piece is better than a 470. On top of that you want sockets too, which you'll get by running a lot of visions.

I think you'll be able to squeeze up to 460 though over time if you don't run these activities.

To further answer your bolded question, mythic + is definitely worth it and the weekly chest reward is the main boost.