I didn't have an answer when writing the question but the dialogue in comments above gave me an "angle of attack".
Let's focus on the obligation to say shema. This is a mitzva (#10 of the 248 positive mitzvot in Rambam’s Sefer HaMitzvot, codified in SA OC 61). As such there is no reason to imagine one can voluntarily refrain from performing it.
But does one have to pursue the mitzva if one isn't awake? An answer from msh210 and a link from unforgettableid led me to a very interesting compendium on Sleep in Halacha by R Aryeh Lebowitz. He brings (section V-D, p. 12) the opinion from R Shlomo Zalman Auerbach (Halichos Shlomo, Tefilla, Miluim #12)
that it is obvious that while a person is sleeping he has no
obligation to do any mitzvos, including shema and davening, because he
is considered like a mentally disabled person (“shoteh”). This exemption from
mitzvos, in Rav Auerbach’s view, includes both מצוות עשה (positive
commandments) and מצוות לא תעש (negative commandments).
To be clear not everyone agrees with R Auerbach but he is a posek of great significance. He explicitly discusses the fact people who sleep are exempt from shema.
R Lebowitz continues
Similarly, if one were sleeping, and the time for davening or shema
arrived, one would not be obligated to wake them up. The sleeping
person is simply not obligated in those mitzvos. However, Rabbi
Auerbach points out, one should wake the person up in order to provide
him with the opportunity to fulfill the mitzvah. In short, while he
will not be faulted for neglecting the mitzvah, the sleeping person
certainly does not get credit for doing the mitzvah. For that alone,
it is worthwhile to wake him up.
I am not clear if this means one can let oneself go to sleep and place oneself in such a situation. R Lebowitz addresses the point in the following way
It may be argued that a sleeping person is not considered like a
shoteh who has no obligation in mitzvos, but like an ones who is
exempt due to circumstances beyond his control. The difference between
the two can be explained as follows: A shoteh is simply not an
intelligent person and therefore never has any relationship with a
mitzvah. An ones on the other hand, may be included in the general
obligation of mitzvos, but is exempt because practically he can’t be
A simple practical example of where these two
analyses diverge is one who went to sleep knowing that he would miss
out on a mitzvah as a result of his sleep. If the sleeping person were
considered a shoteh he cannot be held accountable. If, however, he
were considered an ones, this would be an example of “starting off
negligent, but finishing with an accident beyond his control”, and he
may be held accountable for his failure to set up a system by which he
can be woken up in time for the mitzvah.
After checking with multiple talmidei chahamim, no one believes the above is enough to condone going to sleep without putting an alarm clock. At best, it can allow someone who slept through to make up by saying shmonei esrei twice in the next prayer (tashlumin).