Wouldn't it be critical if the agent botched the domain purchase in weird ways ?

Short of throwaway sites (spam etc) it's hard to imagine skimping time on this specific, mostly painless part.

People are skimping time in every part.

I am watching people who can't code build and deploy dashboards and sites with Claude Code (desktop app - they don't use the CLI), then go cap in hand to developer friends to get it hosted on a domain (rather than some Vercel or whatever URL).

Those people absolutely want to risk letting an agent buy and set up the domain.

This is not necessarily as blindly stupid as you might think. Many of these people know that this workflow is no good for writing code that does anything serious (i.e. storing data for people, taking payments, etc.) but there are a huge number of projects that are just websites, dashboard, data visualisations, etc. with static content and public APIs (Twitter is awash with them) and domains are cheap.

A decent minority of these are even quite cool or interesting.

So a lot of people want to put their vibe-coded weekend project behind a nice domain. Why not?

If the rest of your deployment flow is via the agent, needing to switch over to a different context and open up a browser and login (or create an account) and buy the domain absolutely is a bump in the road.