I can tell you know the topic well, but we're at the same place CPI criticism always come back to: "It's flawed, but the alternatives are worse, don't exist, or in some way not practical."
I know you really really want to make this about the flaws in the alternatives, but I have spent many sentences dealing with the flaws in those in good faith while you have yet to reciprocate at all with any in defense of CPI, either in comparative or absolute terms. Because you know that's a losing game. CPI is cooked, the FRED real wages graph is cooked, your conclusion from it is cooked, and you know it's cooked.
Gold is a market where price moves for reasons unrelated to price level. It was stagnant for decades because rich country consumers value it less and now it's on a run as the US weaponizes the monetary system. It's just not useful to represent inflation over the last 10 years.
Housing is even worse. Unlike dollars and gold, every house is unique and in a different location. It's one of the least useful parts of CPI for the same reason it's a worse deflator on it's own.
I use CPI because it's useful, it exists, and it has less problems than the alternatives on a 10 year chart. We're not comparing to a basket of hand sewn shoes and a horse from the 1820s.