I sifted through a lot of "contact form" type of plugins before I finally found this lovely one. It gives me exactly what I wanted, with a lovely UI. No issues on install or testing. The only non-intuitive change was for the "message on submit", but it turns out if you check the box for edit the text for the other form fields, then you can edit that field as well.
All-in-all, five stars. Exactly what I was looking for!