This will be a partial answer to your insightful question.
This article cites Talmud Brachot 32b that lists the general reason for Psukei Dezimra. In short, it states that one must prepare himself by reciting the praises of G-d prior to making requests from him. While not explained in this article, this concept would explain the reason for the 1st 3 brachot in our Amidah, which focus on G-d's praises, and not our requests from him.
A view beginning at about the middle of page 6 of the article reveals the Ba'al Hatnya's reading which explains that the term tefillah when used in the Talmud generally means the Amidah. However, Ba'al Hatnay states that here, it refers to Kri'at SHema, since the Shema is a mitzvah from the Torah and, according to most opinions, Amidah is Rabbinical. Thus. he says that in order to say the AMidah, one must reach a personal "high point", which is the result of reciting Shema. Yet, one must work his way up to that personal his level by first reciting Psukei Dezimra before Shema to reach that high point gradually.
Well, this would explain why we don't have Psukei Dezimra at mincha as there is no Shema. OTOH, as stated in comments, the recital of Ashrei may be sufficient to accomplish that same "high point" that's mentioned. See the article and the Talmud about the reason for reciting Ashrei in mincha.
Shema is in Ma'ariv, and based on the Baa'al Hatnya's answer, it seems that one should require Psukei Dezimra for Ma'ariv as well. I have to research this a bit more, if I can find something. Offhand, I'm surmising that since there is an opinion that Ma'ariv is Reshut ("optional"), it may not require the same "build up". Or, perhaps the various Birkot Shema themselves accomplish the job.