Rival Executioner draws Robespierre...does this end the day?
Yes, you end the day if you collect Robespierre via the Rival Executioner.

The game only has one verb for the process of operating the guillotine: "collect." When a noble is collected, you do all of the side-effects of that noble. Guillotine is a mix of tactics and luck, so drawing Robespierre due to the Rival Executioner is well in line with the way the game typically plays out.