The citizenship test would be okay, but beyond that...
We *still* can not create standardized tests that are not culturally/socially biased.
A citizenship test is explicitly forbidden by the Fourteenth Amendment, for those born in the United States. Its wording renders the children of illegal immigrants, foreign dignitaries and the like born on US soil citizens, much less you and me. The Fifteenth and Nineteenth Amendments are problematic with respect to testing one's competency to vote. If it's possible to demonstrate systematic racial or gender bias, and it's overwhelmingly likely that it would be irrespective of whether or not it is biased, the courts are likely to strike the requirement.
Your guess is as good as mine what happens if only the Civil Rights Act of 1964 applies (eg: if said test only proves biased against religion/national origin/anything else to which the former has been extended), or how the courts would react to an unbiased test in light of the restrictions on literacy tests and the like enumerated in the Voting Rights Act of 1965. Personally, I would read the intent of the latter as having been to preclude denying the right to vote on the basis of a low level of education, period. The courts might see that one differently. If Congress struck or modified the relevant provisions of the VRA, that's different, but I can't see a majority that passes a competency test altering that wording. There's too much risk involved in places like Florida and Georgia.
Articles I, II and III spell out the requirements for representatives, senators, the president and vice-president and the federal judiciary. We can't change the requirements for Congress or the top two officials in the executive branch without an amendment. We can administer competency tests for executive branch officials however we like, and we do for entry into the civil service. We could run those all the way up to Cabinet level if Congress so desired, and the Supreme Court would be likely to uphold. In principle, we could do the same for the courts, but the Supreme Court will always, always vote to strike that down.
Long story short, the changes most of you want cannot be enacted without a constitutional amendment, which is impossible in this environment. If gerrymandered Republican states want to dismantle their education systems, there is no legal remedy under the Tenth Amendment. The only available option is the use of force, which means a second civil war.
The UK can fix the problem, because they are not bound by the straitjacket that is the Constitution. We might be able to, if we put a majority on the Court that put a stop to the gerrymandering. That's the best argument for packing the Court the next time there's a Democratic majority, but now that Democratic states are just as gerrymandered, I have a hard time seeing a thin House majority voting for it. Too many members in the majority would stand to lose their seats.