You forgot about the part with good hardware. For 3x3 it's true that its the Cuber not the cube but that does not apply to big cubes. Finding hardware you are comfortable with is extremely important to big cube solving. If you want to know my mains in order 4-7 they are Aosu GTS2 MGC MGC Spark. All of them are magnetic and I think that they are the best on the market as of right now. The MGC6 and Spark are without a doubt the best on the market. 4x4 hardware is either the Aosu WRm or GTS2, if you like the size of the Meilong 4x4 get the WRm and if you want something bigger get the GTS2. For 5x5 there is the MGC, Aochuang GTS, and Valk 5. If you want raw speed with a slightly blocky feel and stability get the MGC. If you want something flexible with very smooth turning and corner-cutting get the Aochuang GTS. If you want something with control get the Valk 5.12 things for getting better at big cubes are:
1: Do lots of solves.
2: Doing casual solves were you find out new ways to do things such as centers and edges.
3: Drill parity Algs.
4: Do lots of solves.
5: Make sure your turning is accurate.
6: For 4x4 and 6x6 make sure your centers are in the correct orientation.
7: Get good cubes and/or set them up well.
8: Try to reduce pauses.
9: Do lots of solves.
10: try to plan as much as possible in inspection
11: Maybe try hoya
12: Did i mention do lots of solves?