If we take the website on, the finished site runs on our hosting. That's how we can stand behind the configuration. Packages include a year of managed hosting.
We don't take over your existing infrastructure. ASP.NET on Azure, .NET MVC, a WordPress box, a JavaScript app someone else built — we don't pick that up and try to guarantee it. We don't work in those stacks. Your live site stays where it is, on your infrastructure, while we build a copy on ours (typically Next.js, with a backend where it's needed).
We only switch DNS when you're happy with that copy. Until then, nothing on your side moves. After the switch, we host it. You own the source. If you later want to run it yourself, we can hand over the repo and configs. The default — and the only setup we stand behind — is we host it.