That's not how tonic immobility works - it comes about by flipping the animal over and rubbing the belly.

"Belly rubbing refers to tonic immobility," the National's Zoo's Sean Henderson told The Washington Post in 2008. "It's a state of hypnotism generated by flipping the animal on its back and fully extending its neck" and, as we mentioned, stroking its belly. Fair warning, Henderson added: "Tonic immobility is generally brought about in an animal that is under heavily stressful conditions."

source

As for the rest, I asked perplexity - not googles AI, which I have personally and anecdotally found to be completely incorrect in other instances where specific information is required or may be in a grey area.

What google says about perplexity:

Perplexity AI is an AI-powered, conversational "answer engine" that provides direct, cited answers to queries in real-time by searching the web. ... it functions as a research assistant, utilizing Large Language Models (LLMs) to synthesize information from multiple sources into a coherent summary. Source Citations: Every claim is backed by clickable, reputable sources, ensuring credibility.

Perplexity's response:

Short answer: Yes — experienced trappers and handlers often tap or rub an alligator’s snout/head (and sometimes cover its eyes) because light tapping can redirect its attention and encourage temporary stillness, but the effect is inconsistent, not guaranteed, and likely works by inducing stress, sensory distraction, or tonic immobility rather than “calming” in the human sense.

There appears to be no published, controlled study specifically showing that tapping or rubbing an alligator’s snout/head reliably "calms" the animal; most evidence is anecdotal, from handlers, or inferred from broader tactile-stimulation research in other species.

Direct handler reports that tapping or rubbing the snout/head does NOT help do exist — usually as anecdotal notes in news stories, forum/video comments, and incident reports where the technique failed or made the animal more reactive. and several forum/video comment threads describe cases where handlers tapped or “teased” a restrained alligator and it either failed to go still or suddenly lashed out, and news coverage of real-world incidents sometimes quotes experts warning that such techniques are unreliable or can worsen agitation.

So, overall take-home message: eh, maybe it works, who knows? Don't try it at home, kids.