It is true that there are different kinds of peshat. For example, we have Rashbam's comment (to Gen. 37:2)Rashbam's comment (to Gen. 37:2) that Rashi himself, who aimed at peshat (נתן לב לפרש פשוטו של מקרא) agreed that new peshat-based interpretations are needed (והודה לי שאילו היה לו פנאי היה צריך לעשות פירושים אחרים לפי הפשטות המתחדשים בכל יום).
That said, it is worth highlighting the approach of the Lubavitcher Rebbe zt"l. He takes the view that every detail in Rashi's commentary - even to the extent of which words he uses as the catchphrase (דיבור המתחיל), the citation of the name of the Tanna or Amora who gave a particular explanation, and the order in which he gives multiple explanations - is significant on a peshat level. Each of these, he says, implicitly solves some problem that could occur to a child studying Chumash (who has not yet studied Mishnah or Gemara, so that any problems that he thinks of must emerge from the text itself). Numerous talks of his over a period of some twenty years (1964 to about 1988) are based on this approach, which is summarized in the introduction to Likkutei Sichos, vol. 5, as follows (translation is mine):
"The central idea of these explanations, as the Rebbe often explained, is based on the rule that Rashi himself stresses about his commentary (to Gen. 3:8 and 24): 'I have come only to give the simple meaning.' This means that Rashi's comments primarily set forth the peshat way of learning and understanding the Torah's verses - so that, in the light of his comments, they become understandable even to a child who is first starting to study Chumash (at age five, as set forth in the Mishnah). This peshat approach of Rashi's - and the fact that every word of his leads to the simple meaning of the verse - is thoroughly illuminated by these explanations..."