They push it because it’s an easy win. Corporations will subvert any cause that doesn’t have directly to do with overthrowing capitalism. They do the same thing with environmentalism and diversity initiatives. That doesn’t mean that those things are inherently bad or part of some sort of secret plot.
I guess the question is: what problem are you trying to solve by instituting age limits and term limits?
If the issue is the advantage of incumbency and having entrenched politicians with large campaign funding operations behind them, then maybe a better way of solving this would be campaign finance reform that prevents private dollar donations from non-individuals and heavy restrictions on how much an individual can contribute.
All that term limits and age limits in Congress would achieve is setting an artificial barrier for those who do the job well while setting up a new group of people to benefit from the legislature’s dysfunction.