Acid Accident Treatment
My Lab First Aid book$^{*1}$ tells the following:


  The local therapy mainly consists of the instant, intense rinsing (at least 10 minutes) with tap water (ideally 15 to 20 °C). Don't try to neutralize the chemical burn, because the heat generation could lead to further tissue damages.


This means, that your lab instructor is absolutely right.

The rinsing water should wash the cold water away faster than the heat from the acid base reaction could harm you. It's not for nothing that there are emergency showers and eye douches in every lab.

I have a friend who is a chemical lab assistant and once had to carry concentrated sulfuric acid in a big glass beaker. Suddenly the bottom of the beaker broke off and all the acid poured over her etc. She not even thought of anything else than getting out of her clothes and under the next emergency shower.


$^{*1}$ The book is: R. Rossi, Erste hilfe bei akuten Notfällen - Begleitheft zu Jander - Blasius, S. Hirzel Verlag, 2005, p. 11f.. The text is in german and I translated it.