The problem you encoutered is that the rewritten function was either no longer table-valued, or else it was no longer deterministic (which is what that big list of rules for inlining really means). But that problem doesn't go away by adding a layer of abstraction. The need to understand relational determinism doesn't disappear. The need to understand SARGability doesn't go away. You can't really abstract the problem away.

That being said, CTEs are a really good way to write complex queries. They let you tag bits of query with meaningful names, and each thing you tag is accessible to every CTE after it so you can build up an almost imperative data flow by just doing select transforms one after another. That way you're building hard queries from the bottom up rather than the top down.