Kushner is the point man for the Middle East because Trump's attitude is that professional, full-time diplomats have consistently failed for 40 years, so why not try someone different? One of the prerequisites for such a role, arguably the most important one, is that both sides the envoy is mediating for know that said envoy has the president's trust.
As for his wife, again, there are no formal classes on "hosting meetings at the UN" at the Kennedy School or SIPA or Georgetown SFS, nor is there a "hosting meetings about women's rights at the UN" job description at State or anywhere in the executive branch (the existing US delegation to the UN's responsibilities are different). She either does a good job, or she doesn't and Trump sends someone else. Given that Ivanka has now performed the role over multiple years, apparently her father approves of how she's done.