Is there any indication or good reason to think that the Holy One, blessed be He, joins thought to deed, and that sincere murmuring done with the intention of serving God is favorably received?
This is a follow-up question, but also a general one. For example, if someone is careful about an incorrect prohibition, or puts on invalid tefillin, and the like.

It seems obvious to me that yes. Good intentions are something of value, and I assume that the Holy One, blessed be He, values that as well. The act itself, as such, is worthless. It's like the shepherd's flute-playing in the Baal Shem Tov story. Among Hasidim they go so far as to claim that he fulfilled his obligation of prayer with it. That is, of course, nonsense. But it is definitely an act with value.
In my article on citric acid on Passover, I discussed two passages from responsa of the Ben Ish Hai in which he speaks about the Baghdad community, who had been putting on invalid tefillin for generations, and he argues that they are not in the category of "a head that never wore tefillin," and another similar case.

In Shaarei Teshuvah of Rabbeinu Yonah, he says that every person should study Torah, or at the very least recite praises. From his words it seems that this has at least the value of Torah in the object itself, even if on a low level. Perhaps it is no coincidence that Rabbi Yehudah ben the Rosh was the one who divided the Book of Psalms into weekly and monthly readings (Rabbeinu Yonah's influence on the Rosh and his sons is well known).
In my opinion, there is also a spiritual value of prayer in reciting Psalms, in which a person walks hand in hand with God through the waves and crises of reality (regardless of the question of active or passive individual providence). This creates a bond between the person and God that is not merely the analytical commitment of the rationalist individual who observes the commandments out of gratitude to God. A person lives with God, and by virtue of that cleaves to Torah and the commandments. He loves God, and therefore keeps Torah and the commandments despite the crises and waves with which life floods us. There is something similar here to married life, in which a person's commitment to his wife and her wishes does not stem only from the legal obligation defined in the ketubah. He lives with her, and therefore takes care of all things even if they are not explicitly written in a legal document. (And perhaps this is the source of the pious person in Mesillat Yesharim who does his Creator's will even if it is not explicitly written in Jewish law.)

To the esteemed questioner,
Why casually slander Rabbi Ovadia?

https://www.google.com/amp/s/www.kikar.co.il/_amp/haredim-news/434364