http://www.merriam-webster.com/dictionary/republic
Is illuminating. I particularly like the part about elected representatives and officials being responsible for exercising the power given to them by voters (sound democratic to you?).
Specifically the USA is a Constitutional Republic with the ultimate power derived from the voters. Until recently, I believe this was completely true. But with states like Utah, Wyoming, Colorado, etc. where the common voters were deemed unnecessary, that kinda takes out the intended rights of the voter.
Here’s something to consider: The nominee for president has absolutely no authority to do anything other than run for office.
The stuff you are upset about - it grants no power.