I will always believe, if fitted properly, it is the golfer's swing that makes any club work well for them. Cheap, or expensive, it does not matter. Some clubs might be of better quality than others, but that's about it.
Many years ago, I took a group lesson from a pga instructor. Someone in the group ask the instructor if he was any good. He proceeded to hit balls with clubs from our different individual bags like they were custom fitted to him. He told us, that all he did was make a grip adjustment to offset any difference in our clubs's different lie angles. It did not matter what the brand name on the club was. All the balls he hit, went towards his intended target, at the correct distances. His own swing sent the ball flying. This was the only thing I took away from this lesson. The next night there was a different pga instructor, which is another story.
And fast greens can be near impossible to get the ball close on down hill putts, the margin of distance error is really thin. On my most recent round I had a 60 foot downhill putt that I hit 90 feet and right off the green, I went fringe to fringe. Ended up with 4 putts and the greens were not that fast.
I think it also depends on green conditions too. Firmer greens with significant contour can tend to knock even a steeply descending ball with lower spin far from the landing spot due to the extra energy from the height being deflected laterally by the slope angle at the landing spot (unless the slope acts as a backstop).
A lower trajectory 2-hop & stop shot with lots of spin may tend to hold more contoured, firmer greens better as they meet the slope at a shallower angle with less vertical speed reducing chances for lateral deflection due to the slope angle. The surface friction from spin opposing the horizontal speed of the ball is what holds the ball near the landing spot. If the green is relatively flat then height usually helps even if the greens are firm.
I think it better to practice on as many types of greens as possible . I have noticed that if you don't you only get the feel of one course To be a complete putter need to play slow and fast greens.
Slow green are great to develop confidence on short putts but sometimes can be too bump on long putts
Fast greens roll truer on longer putts and I like to see how close I can the ball and judge fast greens but short putts can be scary as there more chance for lip outs and longer 2nd putt
There is one in the office where I work.
If I use the Five Simple Keys as a framework what does it help:
1 - Steady Head, nope it's not for this
2- Weight forward, nope it's not for this
3 - Flat left wrist (at impact), perhaps this
4- Diagonal Sweetspot Path, I don't think so
5 - Clubface Control, not really, maybe a little but it would be more efficient to just use the either side of the target drill with a regular club and ball
So maybe it would help with key 3? If that is your problem, maybe it will help you. I don't think it's necessary though.