Except people in a typical company who might provide quality feedback on the new website, do not want to "browse Figma". They want an actual website to view - dev site or the actual new live site.
> "You can't just launch a new website and surprise your team."
Yes you can. Nothing beats launching a new site to motivate quality feedback. If you expect feedback to arrive soon after launch, you can use this to your advantage.
If you like the sound of crickets, share a link to Figma and ask people for feedback. Designers and developers will respond, others like sales and non-technical staff often won't. Why? Because people prefer the security of their web browser's familiar reference point when assessing a website. As opposed to browsing an app that gives them a special window to a slippery canvas where an impression of the website is found.
That’s like half the point of prototyping software vs graphic design / illustration software!
Non-designers want to see “the site” open in their browser and be able to click through the pages, instead of use some design app—that’s like half the point of figma.
There's also a lot of work needed up-front on responsive design (as opposed to it being an also-ran or afterthought), and a move away from older typical WordPress-oriented static designs to modern dynamic JAMstack-oriented development methodologies.