Custom domain for landing page validation

Currently in the validation phase. I already bought a custom domain to host my email address, so I can do cold email outreach to get interviews -> to validate.

I also used a free website builder to make a landing page to collect emails but to connect it to my custom domain there is an extra fee. I want to keep my spending as low as possible before this idea is validated. Of course, I plan to pay for it once the idea is validated

So would it be worth it to pay extra to have my landing page have a custom domain, rather than something like "www.framer.startup_name.com". Would this negatively affect how many emails I collect?

Alternatively, are there any good landing page builders which allow you to connect it to a custom domain for free?