The IC leveling at Apple is similar to what you have described. ICT 3/4/5 roughly maps to what you have. The problem with this is that while it's relatively easy to go from a 3 to a 4 after a few years, it's a huge jump from 4 to 5, and then after that you've basically peaked for life. If you just care about building great things, it can work, but for many people, feeling like there's growth and not stagnation is a problem. It also creates a problem for managers having to temper those expectations.
Limiting IC layers doesn't remove all the other politics.
Not really true anymore unfortunately, and the system is converging with other large organizations.
For example, to keep compensation in line with market and reduce attrition, there were definitely more people who got promoted to ICT5 last year and the designation has thus been diluted.
Furthermore, candidates coming in from other large companies expect the title/leveling prestige in many cases, and ICT5 is a tough sell while trying to hire a Google L7/8. So Apple does have a fair number of ICT6 from that.
I also think limiting IC layers and keeping the above politics minimal can work - Netflix did that for a long time and was very successful as a company with their approach. Most IC at Netflix were simply titled "senior software engineer", with pay being a wide band dependent on market value. They no longer do that for some reasons, and have adopted the standard large level hierarchy.
Limiting IC layers doesn't remove all the other politics.