Alternative proof for the undecidability of $A_{TM}$
There is no paradox. We have made an assumption, namely that the halting problem is solved by some Turing machine, and using it reached a paradox. Hence our assumption must be wrong. In the same way, Russell's paradox is a proof that the collection of all sets not containing themselves is a proper class.

In both cases, the paradox is solved by concluding that some object cannot exist. In the case of Russell's paradox, we can give a description of the object - properties it satisfies - but it doesn't exist as a set. In the same way, the halting problem is definable, but there is no Turing machine that accepts it.

In recursion theory we can breathe life into it by making it an oracle, and consider the implications. Similarly, in set theory we can consider relativized versions of Russell's set (relativized to some set "sub-universe"), though due to the axiom of regularity the result isn't interesting.