Don't think I've met any push cart haters, except greens keepers who still can't give me a logical reason why the push cart can't go in between a sand trap and the green.
The Sun Mountain Reflex cart is awesome.
One word, instruction. You don't know what you're doing right to hit those miraculous shots, or what you're doing wrong to hit the lousy ones. You try tips from your buddies, but in all probability they're no better than temporary band-aids, and at worst they're conterproductive. Going to the range on your own, with no idea of what you're trying to work on, is exercise, its not productive practice.
Find a competent instructor (I can't really tell you how to tell a competent one from a bad one), identify a priority, and spend time on the range to work on that priority, and keep repeating that cycle. You can definitely get good advice here by posting videos.
Until someone gives a ruling from the USGA, I'm sticking with the standard ruling that if something isn't prohibited, then it's allowed. Nothing it the rules says that you can't stand on a obstruction that is already in place to play a stroke. A tee marker is nothing more than an obstruction with a specific attribute attached to it which applies only for a finite space of time. That additional attribute has nothing to do with such a prohibition. This is taking 1-4 beyond it's scope.
I also dispute that such an act could ever truly assist anyone in making a better stroke. Very few tee markers I've ever seen would provide any kind of a stable surface. It's hard enough to swing with good balance on a firm level surface. I think if the OP tried this once, he'd quickly give it up.
Step 1. Stop taking tips.
You'll want to build yourself a plan to get from where you are to where you want to be, and piecemeal tips will only detract from that.
Step 2. Find a good resource. You'll want to find out for yourself what the fundamentals of a good swing are.
I am not a salesman, but if I were to start now as you are, I'd purchase the book Lowest Score Wins (http://lowestscorewins.com/) and start there with the intro section on advancing the ball. That's the key- advance it consistently. You'll find lots of tie-ins with that book in this site here, and you can create a "My Swing" thread here (film your swing properly too: https://thesandtrap.com/b/playing_tips/filming_your_swing).
Armed with the book and your swing area, you'll find instructional content below that is cross-referenced a lot. TONS of information in there, all organized by 5 keys:
Step 3. No matter what methodology you chose to guide your learning, stick to fixing one thing at a time. Use slow motion and video to assess yourself. Learn how to practice:
Going to the range and pounding away at balls is WRONG. I've learned that.
Step 4. Have fun. Welcome to TST, and if you choose to improve here, I can say there'll likely be lots of support (particularly if you go the route in Step 2, but not required)- if you choose to interact with other topics here and generally contribute in topics that interest you.