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Some rishonim have a different version of Rashi. They understand that according to Rashi you need to bake the מצה on erev Pesach but even in the morning. This is clearly not connected to קרבן פסח as the קרבן פסח cannot be brought in the morning. The question is if so, why do you need to bake the מצה on erev Pesach?
RYBS explained as follows. The Yerushalmi has a din that it is prohibited to eat מצה on erev Pesach. The Ramban explains this as follows (why is the prohibition limited to erev Pesach). מצה is defined as being not חמץ. Without חמץ there can be no מצה. Before erev Pesach there is no halachic significance to חמץ, it doesn't really exist, therefore there can be no issur to eat מצה as if there is no חמץ there is no מצה. On erev Pesach when the mitzva of תשביתו destroying חמץ starts, then you can have מצה and therefore there is an issur to eat מצה.
Based on this RYBS explained Rashi. We know that מצת מצוה needs to be made לשמה. Rashi holds like this Ramban, that מצה is defined as being not חמץ and that without חמץ there can be no מצה. Therefore, before erev Pesach you cannot bake מצת מצוה because it can't be made לשמה as there is no שם מצה yet. Only on erev Pesach where there is halachic significance to חמץ can there be מצה made לשמה. Therefore Rashi says that the מצה needs to be baked on erev Pesach, even in the morning, to create the לשמה of מצת מצוה, and מצה baked earlier is no good because it lacks לשמה.
This also explains why they would bake the מצה on the seder night if erev Pesach was Shabbos.