Q:

Why not cook a z'roa just for the seder plate, in light of mitoch?

It's customary to have a bone (with some meat on it) on the table for the Passover seder. Kitzur Shulchan Aruch 118:5 (as cited in the answers to another question on this site) writes:

Once the Temple was destroyed, the rabbis established that, at the time the hagada is recited [at the Passover seder], two types of cooked food be on the table…. People have practiced that one of the cooked foods is meat, from the limb called z'roa….One must roast or cook them [before the holiday starts]. If he forgot…, he should roast or cook them at night, but must eat them the first day of the holiday. [The "day" runs from nightfall to nightfall.]  Likewise, for the second night [seder], he should roast or cook them and eat them [all] on the second day of the holiday. This is because we do not cook on [a day of] yom tov for another, nor on yom tov for a weekday. (And, since we don't eat roasted meat these two nights, he has to eat the z'roa by day specifically.)

In other words: One can cook on yom tov, but only for the use of that same day of yom tov. So don't cook the bone on the first night of Passover unless you'll eat it the first day.
But I'd think that one could cook it the first night of Passover, and rely (for "use it the same day" purposes) on the placement on the table in fulfillment of the custom to do so. That itself would seem to be a use. Indeed, when KSA itself codifies the rule that cooking — and likewise lighting a fire — on a holiday can be for the same day only, it says (101:2):

Likewise, it is forbidden to place the lamps in the candelabrum… on the first day of the holiday for the [following] night, unless he needs to use them before nighttime also or for the honor of the synagogue.

This means that lighting candles in a synagogue, even by day when the light is not needed, is a perfectly good use of lighting a fire on the holiday, since it's for the honor of the synagogue. (See also KSA 13:1.) And in general (though I cannot find this in KSA) we say cooking and lighting are permitted for any activity necessary for yom tov, not only for eating (a principle called "mitoch").
Nonetheless, displaying a cooked bone at the seder is not a sufficient use to permit cooking on the holiday. Why not?

A:

The Rema at very beginning of Hilchos Yom Tov (495, 1) rules that the permission to cook food on Yom Tov because of 'ochel nefesh' only applies if by cooking the food before Yom Tov the taste will deteriorate by the time you eat it. However, if your food will still taste fresh, one should cook it before Yom Tov. 
The Yaavetz in his Siddur vol. 2, p.20b, ten lines from the bottom says that because one is not going to eat the 'Zeroah' at the meal one has to roast it before Yom Tov, and the same applies to the egg even if one will eat it that night, since the taste will not worsen if one cooks/roasts it before Yom Tov. Clearly, if one isn't allowed to cook these items on Yom Tov even if one will eat them, all the more so one isn’t be allowed to cook them just in order to display them.

