Have you heard the famous story of the Vilna Gaon and the etrog?
There was a shortage in Vilna, and the people of the town were able to
locate a single man in possession of an etrog befitting the Gaon, but
he wouldn't part with it for any price. He did say, though, that he
would part with it if the Gaon would give all of the reward he got for
the mitzva to him. When the Vilna Gaon heard this, he was overjoyed,
and danced with more fervour that Succot than any other year, claiming
that he had waited his whole life for an opportunity to perform a
mitzva completely Lishma, completely for Heaven.
Prayer is the main avodah, service1, and there are three levels to this service: with our heart, our soul and our might2. As you've pointed out in the question, and a lot of people would express similar sentiments, that we don't connect to the Tehillim, meaning our heart isn't in it. Even though we are aware of the reward and the fact that it fixes the world, this also doesn't seem to inspire us; our soul isn't in it either. So we are left with our might.
If we are not thinking of the reward, or following our heart, then we are simply doing it for Him. No ulterior motive, and that is our might, and indeed sometimes we need to not be in the mood to expose that diamond in us, which is that we truly do it for our Master, not our personal gain. It has been said3 that our generation only has access to this type of service, which corresponds to the highest service of all, the service of the ben aliyah4, who is motivated not at all by his own gain or fulfilment, but by the fact that this what his Master needs of him5, and is therefore doing it Lishma6.
The only thing many people of this "lowly" generation need to know is, is this truly needed of me, or not. Indeed, it is. In the leshem yichud we say before reciting Tehillim (e.g.), we explain what the purpose of saying them is: to unite the Holy One Above with the Shechina down here, we aren't doing it for reward or ourselves at all*, we are doing it for Him7. This particular job, the Lubavitcher Rebbe says, is tzorchei gvoha8, the needs of Heaven, and it is צרך גמור הוא, absolutely essential9. Of course it is, Torah is essential.
A mentsch is someone who, even though he doesn't necessarily understand why he should, nor sees personal gain in doing so, and even at times when he is not even in the mood, is always happy to give his family, community and God what they need; tell them the words and praises that they need to hear, and that his essential relationship with them needs in order to be thrive in holiness and health. To take this idea from general to specifically about Tehillim, we can now say that Hashem waits expectantly for us to say His words, the heartfelt poetry written by His beloved people (the King represents all of the nation10), which are filled with many loving praises and secrets. This invites Hashem down to His Shechina, His dwelling with His Reishit, Yisrael, the purpose of creation11.
Serving with one's might gives one a hard-earned opportunity to get away from worrying about one's owns needs and truly invest in serving someone else, and saying Tehillim truly is one of the key, essential services of our beloved God1,12. May we be blessed that in doing so, we find the promised simcha13 that is found in our might, in the joy of giving to others what they need (just because we are lowly, does not mean we do not have a real mentsch on the inside - based on the works of the Arizal, especially Shaar HaGilgulim), and that simcha should poretz gedarim, and bring our heart and our soul back in to our service as well, which will create a momentum to learn more, and delve more into the general and specific themes, commentaries and secrets of Tehillim14, and give us a tayva to sing them joyously each day.
I hope this helps discover our own connection and fire in saying Tehillim.
tl;dr: Sometimes, the only reason we are not in the mood is because we are feeling it is about us, our segula etc, and realising otherwise is all it takes to feel at home again.
* This is a Divine relationship of "I am for my Beloved, and My Beloved is for me" (שיר השירים ו׳:ג) where we both seek this dwelling, and indeed we will find that He will look after us, for instance we will find a lot of timeless relevance to our lives in the Tehillim, and those words will become prayers that He answers, B'EH!
1 - משנה תורה, הלכות תפילה וברכת כהנים א׳:א
2 - דברים ו׳:ה
3 - For example.
4 - Tanya Chapter 10
5 - As in משנה אבות א׳:ג
6 - See Rambam on ibid; משנה תורה, הלכות תשובה י׳:ד-ה; מסילת ישרים י״ח
7 - הסולם על ספר הזהר א׳ כב
8 - בשעה שהקדימו, יום ב' דחג השבועות, ה'תשי"ב
9 - לא תהי' משכלה, ש"פ משפטים, פרשת שקלים, מבה"ח אדר, ה'תשי"ב
10 - שלבו הוא לב כל קהל ישראל, משנה תורה, הלכות מלכים ומלחמות ג׳:ו; תלי בעולם כמלך על כסאו גלגל בשנה כמלך במדינה לב בנפש כמלך במלחמה, ספר יצירה, פרק ו
11 - רש"י על בראשית א׳:א׳; See also footnote 8; Based on various midrashim, cited in Tanya Likkutei Amarim, explanations there.
12 - ואהבת את ה׳ אלקיך
13 - Based on תהילים ק׳:ב
14 - I found this hidden gem of a comment, with a wide spectrum of sources to pursue, in the yeshiva world coffee room topic on this same question
