I'm pretty sure you don't get "make X vote for us" missions as a republic.
Right, I realized you wouldn't get 'Get a royal marriage with X', but didn't apply the same reasoning for the HRE missions
Still, I think "Well, as Switzerland you get a lot of conquest and vasallisation missions" gives a wrong impression, seems it's a general recommendation based on your personal stroke of luck you had with them.
Start with some fairly wealthy provinces and easy pickings up north until you start to run into the Arabian nations and Mamalukes, if you concentrate on your sliders you should be able to get rid of the tech issues with westernisation.
Am in the end game of a Brunei > Malaya game where I've kicked the Europeans out of SE Asia and India. I own all East India trade ports and all of the Spice Islands. Westernised to become tech leader and am now stretching my advantage with massive trade Empire.