As a non-scientist I've always found the Cosmic Distance Ladder as likely to be inaccurate due its assumption about the constant brightness of Standard Candle stars over their lifetime, and the compounding of error at each rung of the ladder. Direct measurement of the CMB seems to be simpler with less chance of error.
Direct measurement of the CMB can also have problems if our assumptions about it are wrong. A major goal of having two methods is that they should coalesce to the same result within margin of error - that they didn't told us we were missing something.
> I've always found the Cosmic Distance Ladder as likely to be inaccurate due its assumption about the constant brightness of Standard Candle stars over their lifetime
Stars are just basic nuclear physics and gravity, that's why they're expected to be stable and predictable.
> Direct measurement of the CMB seems to be simpler with less chance of error.
Direct measurement of the CMB doesn't tell you anything on its own, you have to interpret the data in terms of a model. If you have a completely different model, say one without dark energy or without dark matter, CMB measurements would tell you something different than LCDM.
The supernovas type Ia luminosity depends on their composition and that takes into account both the age of the supernova and of the donator star. And that can be inferred by the luminosity curve of the supernova.
https://en.wikipedia.org/wiki/Cosmic_distance_ladder