Why is it only electrons that take part in chemical reactions?
The nature of an electron is more complex than a single charged particle. Once you get down to the quantum level, our intuition about how matter interacts with other matter fails.

A simple answer to your question is that electrons occupy a much larger volume than nuclei. When two atoms get in close proximity to one another, their electrons will "bump into one another" and interact far before their nuclei will. 

You are also correct that the nucleus of a neighboring atom is essential for the electrons to interact with one another. If there was no positive charge to attract another electron, a bond would not be formed. The point your instructor was trying to get across is that in most (excluding nuclear reactions) chemical reactions, electrons are the species moving around and forming/breaking bonds.

If you would like a more in-depth discussion on the true quantum nature of electrons and their interactions, you should check out Mulliken's Nobel Lecture from 1966. It is a great introduction.