...believes in the concept of 'weapons exist to supplement powers' and not the other way around on the vast majority of characters?
They exist to support each other. However the situation is a bit asymmetric. You can finish a gold/plat game easily with only weapons never using a power to cast, yet still get a (very) good time. On the other hand using powers only is viable on bronze/silver, and with a notable exceptions of a few powers (cain mines, flamer, huntress) it takes a long time to take down gold/plat bosses with only powers. In other words powers do not sclae up as well as weapons. Which means that it is possible not to use your gun but it will draw out the game considerably.
BTW, that is why your scores are so low ...
I feel like everyone here thinks that if you don't work heavily in weapons (and thus the weapons section in passives), you can't succeed. It is quite often that I go a match without firing a shot with a weapon, and go through as a biotic, tech, or melee wrecking ball. I feel like if I rely heavily on powers, I am a bigger contributor to the team effort rather than running off to hit things with a black widow or claymore...
You have nothing to lose to use your gun between casting powers. You are basically wasting time not doing anything. The acolyte is perfect weapon for casters: charge it up, fire, use power, charge it up while the power is on cooldown, repeat. You can basically double(tripple) your DPS.
Also not using your gun is not "teamplay" it is "teamgimp". Your team wants the enemy dead as fats as possible, so use your gun.