Q. Is the usage of “kindly” in “We would like to kindly ask you . . .” or “This is to kindly remind you to . . .” correct? My understanding is that “kindly” sounds rather rude here, as we are describing ourselves as kind and generous.
A. According to the entry for “kindly” in Garner’s Modern English Usage, 5th ed. (Oxford, 2022), using that word to refer to the person making the request instead of simply using it like please is a “linguistic misstep,” one that’s typical of “airlinese”—as in “We kindly ask you to take your seats” instead of “Kindly take your seats.”
Misstep or not, it can be rude to praise oneself, as you suggest, so we agree that it’s best to avoid kindly except as a substitute for please. Your sentences could be revised as follows:
We would like to kindly ask you . . .
becomes
Kindly . . .
This is to kindly remind you to . . .
becomes
Please remember to . . .
(To any fans of airlinese out there, kindly accept our most humble apologies for this answer.)