Two things, one word
Almost every argument about cuticles is people using the word for two different tissues.
The cuticle proper is the living fold of skin at the base of the nail. It is a seal between the outside world and the matrix where the nail is made, and cutting it opens a route to infection. An infection at the matrix is how a nail becomes permanently ridged. Nothing here touches it.
The pterygium is a thin translucent layer of dead tissue that grows out from under that fold and adheres to the nail plate. It is not living, it does not bleed, and it is what makes a base line look fuzzy and what gel lifts away from. Removing it is the whole job.
A soak-and-clip manicure conflates them: water softens both, and nippers take the dead film and a little of the living seal with it. That stings occasionally, and cuticles grow back thicker.
Does it hurt?
No, and it should not feel like anything much at all. A fine bit at low speed on dead tissue is a faint buzzing. If it stings, something is on living skin and the correct response is to say so immediately.
The first one occasionally feels strange because most people have only had the soaked version and are braced for the clip.
Who should skip it
Anyone with eczema or psoriasis around the nail fold during a flare. Anyone on blood thinners, where we do a modified version. And anyone who has had a manicure infection recently, where the honest answer is to leave the area entirely alone for a few weeks.