It is very difficult to prove a negative. This is true both because there always exists the possibility of a small effect that a given study is underpowered to find, and because of the chronic issue with reporting negative results in biomedical literature and science in general (that is, experiments with a negative result are less likely to be published).
However, there is no recommendation for or substantial scientific evidence for sleep benefits of morning caffeine consumption and acute avoidance of caffeine, regardless of time of day, has been observed to have a modest improvement on sleep.
Given the ubiquity of caffeine in society, the multitude of studies concerning caffeine, and the existence of studies that show that caffeine avoidance is better for sleep, I think there is a pretty solid body evidence that suggests any sleep-beneficial role of caffeine is either non-existent extremely minimal.
Several authors have noted that peoples' perceptions of the impact of caffeine on sleep and wake are substantially exaggerated. Caffeine in regular users has little impact on sleep, and the immediate cognitive benefits of caffeine are mostly explainable by a reduction in baseline abilities under caffeine withdrawal. Similarly, sleep can improve in long-term users of caffeine when caffeine is withdrawn, but the effect is both small and short-lived.
I think it is important to recognize that adenosine is not the only signal contributing to sleep pressure, and so it is problematic to treat the system like it is. Habitual caffeine users adapt to the caffeine intake by increasing adenosine receptor expression, but they may also adapt to the increased adenosine receptor expression with modification of other contributors to sleep.
In the individuals of most concern: those with chronic insomnia or other sleep problems, their issues seem to persist despite the inherent increases in sleep pressure that people who lack sleep will accumulate, so there is unlikely to be a "quick fix." The standard practice is an overall emphasis on "sleep hygiene" including setting a regular sleep-wake cycle, avoiding sleep-influencing substances including caffeine and alcohol, keep a standard and comfortable sleep environment, and exercise regularly: however, each of these strategies individually have only a modest impact.
There is also the issue of individual differences: if someone sleeps better in the evening when they wake up with caffeine, then there is little reason to change. There just isn't any scientific evidence that suggests prescribing morning caffeine as a sleep aid.
James, J. E. (1998). Acute and chronic effects of caffeine on performance, mood, headache, and sleep. Neuropsychobiology, 38(1), 32-41.
Nehlig, A. (2010). Is caffeine a cognitive enhancer?. Journal of Alzheimer's Disease, 20(s1), S85-S94.
Sin, C. W., Ho, J. S., & Chung, J. W. (2009). Systematic review on the effectiveness of caffeine abstinence on the quality of sleep. Journal of Clinical Nursing, 18(1), 13-21.
Stepanski, E. J., & Wyatt, J. K. (2003). Use of sleep hygiene in the treatment of insomnia. Sleep medicine reviews, 7(3), 215-225.