Although the Republican leadership’s proposed rules were largely a copy-paste of the rules for the 118th Congress, there are some changes. One major difference has to do with the procedure for removing the Speaker, which played a key role in the 118th Congress. Previously, any single member of Congress could bring up a resolution to remove the Speaker. Now, nine Republicans have to co-sponsor such a resolution for it to be considered. This strengthens the hand of the Speaker significantly and also undermines the ability of a majority of the House of Representatives to change who is serving as Speaker.
It’s been a long time for most of us, but we did have to provide some proof when we first registered to vote. I believe it was having a social security number, which itself has proof requirements. Most of us got our SSN at birth.
Yes, there are some holes in this, but they are very rare and limited.
As for proving citizenship currently, the eligibility documents for a passport are published. Most people have the supporting docs to get one.