If one wanted an axiomatisation of set/class theory in which cardinals became tame and "boring" by imposing an axiom that made all cardinals act "essentially" like the infinities we are familiar with $\omega$ and $\aleph_1$ what is the best choice of axiom.
Obviously GCH is a candidate because it enables us to evaluate cardinal exponentiation. But even GCH is a bit weak when it comes to taming the infinite cardinals, for example we still have regular and singular cardinals, and cofinality still plays a role in calculations.
My question is has anyone explored stronger axioms than GCH which are still "natural" and which further constrain the cardinals. In particular if we took $cf(\kappa)=\kappa$ for every infinite ordinal would that imply GCH and eliminate cofinality as a key concept from set theory.
$cf(\kappa)=\kappa$ for every ordinal (even every limit ordinal) is clearly false - think about $\omega+\omega$.
Even if we look at cardinals, it's still false: consider $\aleph_\omega$, the limit of the sequence $\aleph_0,\aleph_1, \aleph_2, ...$. This is uncountable but has cofinality $\omega$. The existence of $\aleph_\omega$ is provable in ZFC, the key being the axiom of replacement.
So in a sense yes, your proposal would eliminate cofinality as a key concept from set theory.
In order to have a good set theory, you need to have singular cardinals. The fact that they have complicated properties should be viewed as a positive, not a negative: the universe of sets is richer for having singular cardinals in it.
Also, if we're only willing to talk about the $\aleph_n$s then we'll lose theorems like "If $\aleph_\omega$ is a strong limit cardinal then $2^{\aleph_\omega}<\aleph_{\omega_4}$." Life just wouldn't be the same.