Why can’t my website designer purchase my domain for me?
Whoever owns the account that the domain is purchased in, owns that domain. If you let someone purchase it for you , in their account, they own the domain.
I have seen this happen often. It never ends well. Over time the person that purchased your domain might leave the industry, leave the country, retire, or pass away. Maybe they just stop returning your calls or emails. Now what?
If you do not have access to your domain, you will be rebuilding your website in the future on another domain. Not ideal, but I have seen it happen many times.
Protect yourself and buy your own domain, in your account.