As I pointed out in July, illegal immigrants distort our political system even when they don't vote illegally.
U.S. states with large numbers of undocumented immigrants could receive additional seats in Congress after the 2010 census is conducted.A University of Connecticut study concluded Arizona, Texas and Florida could all see their House delegations increase due to rising populations that include sizable numbers of illegal immigrants.
Although they can’t vote, such aliens are included in the census. The San Jose (Calif.) Mercury News predicted Tuesday the pending 2010 headcount could be the subject of a political fight as Democrats and Republicans jockey for position before House seats are reallocated.
Expect to hear more about this now that illegal immigration is a national issue and we're getting closer to reapportionment.
Compare and contrast with your thought experiment on the 19th amendment. Illegal immigrants today are no more distortive, and actually quite a bit less so, than the political system of the US prior to the 19th amendment, or even the 26th amendment.
TML: Illegal immigrants tend to have a totally different culture than Americans, unlike men and women of the 19th century. There's a lot more going on with illegal immigration than electoral distortion.