This is a fundamental that is that ruins so many golf swings before they start. Let's look at why. Bending at the waist is physically easier - it takes less work from your legs to bend over and your body naturally avoids work. The big problem is that you are out of balance by the time you reach the top of your backswing and worse... Your upper body weight is not over your legs which means a ground up swing is dead. Through your transition your weight needs to be firmly planted over your loaded legs so 1) you can linger in your transition in balance until your forward weight shift happens and 2) So that your lower legs are in a position of leverage to get your weight moving and to have enough spring to keep pushing until your weight has made it all the way to your left foot.
So, if you don't love your swing check for this setup flaw. So, many people who swing over the top don't realize they had no chance way back from their setup.
I voted breaking 80 is harder. Obviously the two are connected but if push came to shove I think more golfers could learn how to avoid making doubles than be taught or acquire the skills to break 80. Breaking 80 for many golfers would mean significantly raising the level of their games, while not making a double is more about avoiding the big miss and game planning/course management. More specific to the OP's question relating to a golfers that shoots in the low 80's and is trying to break through, it could be one or other, most likely a combo of both.
For myself, like @The Recreational Golfer I've had more rounds in the 70's than double bogey free rounds so could you say it's "easier" for me to shoot in the 70's than avoid making a double? Yes and no, depends on how you look at it. Even though I messed up on one or two individual holes I had to play fairly solid golf the other 16/17 holes to break 80, IMO that's more difficult than avoiding making a double. Even on a round where I didn't make a double I had to perform at a consistent level for 18 holes to break 80.
If you told me I had to play 10 rounds where I had to break 80 and 10 rounds where I couldn't make a double, I think the latter would be easier for me.
Good to hear. An observation for you... When you start getting your lower body into it guess what? You start doing work! And that takes fitness. Some people have an illusion of an effortless golf swing. But in reality, it's effortless for your wrists but more work for your legs and core. So, keep is mind as you go from hitter to swinger your fitness, core, and lower body strength demands go up.
I'll answer your question with another question:
If you drop the 3H, what would you hit for a 215 yard shot, or even a 210 or 205 yard shot? Are you comfortable gripping down on your 4-wood to cut distance off of it by that much?
I personally don't see much issue with keeping the 3H in the bag since it isn't actually hurting anything. You say the other alternative is to drop the 4-wood, but why do you feel that you must drop anything at all? It's okay to have a 10 yard gap. If it really concerns you, though, I would think that the best way to go would be to swap from a 4-wood to a 3-wood. This would increase the gap between your 3H and the next longer club without affecting the gap between the 3 and 4 hybrid, which seems to be a number you like better for a gap.
I'm just a little confused why you'd drop down to 13 clubs if the hybrid isn't costing you strokes. I know I find hybrids easier to hit than a wood that I'm trying to hit shorter than usual. I absolutely hate to hit a 3/4 3-wood to cover the cap between a full 3-wood and my 3-iron, but the number for a 3/4 3-wood is such a rare distance that I'd much rather have consistent gaps up through a 3-iron than a big jump from a 4i to a 3H. In my case a 3H would hurt my game, but it doesn't seem like yours is doing you any harm and could be protecting you from an awkward 4 wood shot.
This might not be the most helpful for balls and gloves (though I do see golf ball deals pop up at least once a week), but golfwrx's classified section is a great place to get deals on a lot of stuff. I got my AD-DI shaft there for $130, along with my KBS shafts from 3-LW for another $150 for the set. The AD-DI was used for a week (didn't have a mark) and the KBS shafts clearly only ever had a grip installed on the 7-iron. Good deals there, at least from what I've found, but mostly on clubs and shafts (though they do have golf balls in their classifieds every so often).
The nice part about that is they also have a WTB forum, where you post what you want and people come to you. That's actually what I did for the iron shafts. I posted about what I wanted and I had about 7 responses to choose from within 24 hours, so it's pretty active. For golf balls other than ProV1's that would be your best bet for finding them.