This claim is oversimplified/too generalized. The Overwhelmed Brain speaks on love and vulnerability, highlighting how vulnerability can create a sense of security and safety and ultimately lead to a stronger connection, bond, love, and trust. Vulnerability also leads to healing from past traumas and growing with your partner. That being said, the article is more of an opinion piece and based on personal experiences than it is a factual/scientific confirmation of whether the claim is true. Yes, vulnerability can be argued as an important component to developing feelings for another, but the idea that most people fall in love during times of vulnerability is too generalized. Vulnerability is not the only factor when most people fall in love.
There are scientific components that are essential for an individual to fall in love. Health Matters explores how your brain and five senses can help you fall in love, interviewing Dr. Philip Stieg, neurosurgeon-in-chief at NewYork-Presbyterian/Weill Cornell Medical Center and director of the Weill Cornell Medicine Brain and Spine Center. "When people fall in love, there's a lot happening at the neurochemical level." Dr. Stieg further elaborates on how sight, touch, smell, taste, and sound can all contribute to falling in love. For instance, when we see someone we consider beautiful, our brain triggers feelings of lust, with both sexes experiencing higher levels of dopamine and lower levels of serotonin. When we touch someone we are attracted to, whether that be holding hands or a gentle kiss, our brain releases oxytocin and dopamine, as touching is "capable of evoking strong feelings of pleasure." Ultimately, our brains and five senses play a role when it comes to falling in love, not just shared vulnerable moments.
Missouri Partners in Prevention also addresses other factors that influence the likelihood of falling in love, such as similarity in personality traits and patterns in thinking, desirability based on personal preferences and standards, close proximity, reciprocated feelings, and social influences. The source also lists the chemical steps to falling in love: lust, attraction, and attachment. The website includes sources from a Harvard Medical School study, Love and the Brain and a Psychology Today article, The Science Behind Falling in Love.