Thanks for the tip, Terra! Yeah, I'll definitely stock up.
The one thing that makes at least the main eight stages easier is being able to buy E-Tanks and lives in between stages. After getting the Jewel Satellite, and thanks to TerraEsperZ mentioning its infinite usage, there were a couple places I found I could exploit it to gain Screws (and lives) easily without touching the controller. Infinite loops FTW! This will hurt my chances of finishing the game fast enough to get at least that one achievement, though.
In Plug Man's stage, after the disappearing blocks and towards the end of that room are two pipes facing right that continuously spawn those little round robots. By using the Jewel Satellite when standing at the bottom beneath the pipes' openings, the enemies just keep flowing downward to their down and you can reap their items. (If you take a step to the left to bring the left pipe in view, it's not as effective because the left pipe is farther, and the angle they come in make the items they drop get stuck at the pipe.)
In Hornet Man's stage, there are some robots that resemble potted plants that continuously shoot plant missiles at you. I stood to the right of a group of (at least) three of them near the end of the stage and found that with the Jewel Satellite on, two of them will shoot missiles that home in such a way that they are eliminated directly above Mega Man so he can once again build up items easily. (I found that if I shifted to the left, ie. right up against the short wall/ledge/step, the third one over will also shoot missiles, but then you can't get its items because they end up on that very ledge, and you also miss out on the items from the closest robot pot. So it's best to go for two.)
There are probably other places this can be done, but those are the two I came across. Hopefully I can find one near the beginning of a stage that are at least as fast, but then again I'm probably not going to get a good time anyway...and it doesn't take that long to get through the stages, and I shouldn't need to do this too many more times at this point... I will have to revisit those areas to max out on Screws before tackling Wily's fortress.