I was recently told that my website should be responsive to placate those mobile users out there.

Responsive? Sod that.

The internet was created by people who look like this (the lovely folks at ARPANET):

darpa

 

for people who look like this:

dennis-nedry

 

to use on things that look like this:

vt220

 

Do they in any way convey a sense of responsiveness?

Nope.