"אַשְׁרֵי, יוֹשְׁבֵי בֵיתֶךָ-- עוֹד, יְהַלְלוּךָ סֶּלָה" -- Psalms 84:5
"אַשְׁרֵי הָעָם, שֶׁכָּכָה לּוֹ: אַשְׁרֵי הָעָם, שֱׁיְהוָה אֱלֹהָיו." -- Psalms 144:15
The Mishna (Berachot 30B) tells us that the "Early Pious Ones" would wait an hour before prayer in order to properly prepare to pray. The Talmud, (Berachot 32B) tells us the source for this is Tehillim 84:5, "Happy are they that dwell in Thy house, they are ever praising Thee. Selah". First they dwell in your house, and then they praise you (Rashi).
Tosafot (32B) says that this is why the Rabbis established that we say the verse "Ashrei Yoshvei" before starting Psalm 145, to exclude who say many verses that start with the word Ashrei. (In other words, the Rabbis said to say this verse, since it has a special connection to the beginning of prayer, more so than other verses that start with Ashrei)
The Tzelota D'Avraham points out that even in the time of the Talmud the prayer of Tefilla LeDavid was referred to as Ashrei. In Berachot 4B, R' Yochanan asks, if the prayer is alphabetical, why is there no Nun in Ashrei? From here we see that even in the time of the Talmud they were already referring to the prayer as Ashrei.
The Tzelota D'Avraham puts both these ideas together and says that at least as far back as the time of the Talmud, the custom was to say the last verse of Psalm 144 (verse 15) before reciting Psalm 145, since they are contiguous and also thematically similar. (Similarly to the way we make sure to add "Emet" at the end of Shema, because there is a verse that says "V'Hashem Elo-eichem Emet".)
"...happy is the people whose God is the LORD...I will extol Thee, my God, O King; and I will bless Thy name for ever and ever."
Later on (as Tosafot said), they added Psalm 84:5, because of the connection to the beginning of prayer.
The Tzelota L'Avraham later offers an additional answer. He says that they didn't want to start with Tehilla L'David, because it can be misinterpreted to mean "A Song of Praise to David" (instead of "By David"). So they decided to start from the previous verse. However, that previous verse doesn't make sense on it's own, because it says "Happy is the people that is in such a case.". What is the case? So they added another verse before it, "Happy are they that dwell in Thy house, they are ever praising Thee...".
Happy is the people that such is the case, that they are ever praising You.
There is one question we have left. Ashrei is not in the beginning of the prayers, so why would the Rabbis add a verse referring to pre-prayer preparations in the middle of prayer? To this, My Prayer answers that it was originally inserted in the beginning of the Afternoon service. Once they added it there they added it to all the rest of times the Ashrei prayer is mentioned as well.
As for why the prayer books you examined didn't have these sources, I don't know. The Siddur Tehillat Hashem from Kehot (a Chabad Siddur) is full of footnotes. For example, here's the page Ashrei is on.