I am unaware of any medical study that indicates a drop of alcohol is sufficient to trigger a relapse on its own. Relapse is a complex phenomenon.
The quote you provide sounds like it comes from a "12-step" program or similar that considers alcohol as something that is incredibly powerful over addicts and must be completely avoided. This approach works for some but may not be necessary for everyone, and a proportion of people with an alcohol use disorder are able to eventually modify their behavior to drink moderately.
Cooking with alcoholic beverages does not actually remove all ethanol. That said, the amount consumed is unlikely to be sufficient to be detected if someone is unaware their food contains alcohol. Rather, any impact would be psychological. Someone sober who unknowingly ingests some food containing alcohol has not violated their sobriety in any meaningful way.
The most often described triggers for alcohol relapse (after initial withdrawal) involve situational or emotional triggers. Addictions to alcohol or other substances involve changes in brain circuitry that associate reward with those substances; these are similar to the networks involved in seeking food or other sorts of pleasure but are driven very strongly by drugs of abuse.
Situational triggers can include being in locations or with people that are associated with drinking: being in a bar, for example. Tasting a small quantity of alcohol could be indeed be a trigger for some people, but there is no reason to consider those small amounts to be any stronger as a cue than other situational cues. It is unlikely that any one event will trigger relapse on its own, rather, relapse is thought to involve combinations of factors that ultimately overwhelm coping strategies.
Urges to tempt or test sobriety are often associated with relapse, so if someone is tempted to 'cheat' their sobriety through seeking foods that contain alcohol or alcohol-related flavors they may be closer to relapse than someone who simply would prefer not to monitor whether food could contain some small non-intoxicating amount of ethanol.
Daley, D. C. (1987). Relapse prevention with substance abusers: Clinical issues and myths. Social Work, 32(2), 138-142.
Grüsser, S. M., Wrase, J., Klein, S., Hermann, D., Smolka, M. N., Ruf, M., ... & Heinz, A. (2004). Cue-induced activation of the striatum and medial prefrontal cortex is associated with subsequent relapse in abstinent alcoholics. Psychopharmacology, 175(3), 296-302.
Larimer, M. E., Palmer, R. S., & Marlatt, G. A. (1999). Relapse prevention an overview of Marlatt's cognitive-behavioral model. Alcohol Research & Health, 23(2), 151-151.
Marlatt, G. A. (1996). Taxonomy of high‐risk situations for alcohol relapse: evolution and development of a cognitive-behavioral model. Addiction, 91(12s1), 37-50.
Witkiewitz, K., Roos, C. R., Pearson, M. R., Hallgren, K. A., Maisto, S. A., Kirouac, M., ... & Tonigan, J. S. (2016). How much is too much? Patterns of drinking during alcohol treatment and associations with post-treatment outcomes across three alcohol clinical trials. Journal of studies on alcohol and drugs, 78(1), 59-69.