Flour and water are how you create gluten, the network of proteins that gives pasta its stretchy texture and bite. Adding yolks, which contain about 48% water, 17% protein, and around 33% fat, gives the pasta more colour, richness, binding and silkier noodles. If you lose the eggs, you lose the character they give the pasta, which cannot be replaced by the qualities of yoghurt.
While handmade fresh pasta all'uovo all contains eggs, most dried and commercially available pasta is made without eggs, including spaghetti and penne.
That said, if you do want to harness the tangy notes of yoghurt with pasta, you could always add yoghurt and eggs to your pasta mix, as per the recipe found in The Everyday Fermentation Handbook, which Food Republic reports: "adds a mild tart flavour and a slight creaminess that doesn’t fall apart when cooked".
Or, you can even try making pasta with a yoghurt sauce, like this recipe for creamy Greek yoghurt pasta from The Kitchn, which they say makes the ultimate velvety and tangy sauce.