The thing that’s “equivalent to Arch” would be Arch. There is really no comparison between Gentoo and Arch outside of the “build your own system” approach; they use different package managers, different init systems (by default), etc.
Arch has the goal of providing user flexibility through its minimal nature — you’re expected to choose the software you want and build up a system that fits your needs.
Gentoo, meanwhile, has the goal of providing user flexibility through its minimal nature as well as through skillful use of compiler flags and machine-specific hardware tweaks — you are quite literally expected to build your system, as in compile your software from source. Even the installation is like that: you grab a root tarball and do a good ol’-fashioned chroot installation, then edit /etc/portage/make.conf and let 'er rip.
Now, Arch has the ability to offer a certain amount of this flexibility as well; most people become familiar with it by diving deeper into the AUR than just installing yay or paru. With Gentoo you have portage and ebuilds baked into the system from the start. You can have complicated setups involving multiple build machines that can combine resources and have it all handled with some basic mucking around in /etc/portage. Gentoo also offers the concept of overlays, which add repositories like GURU that fill similar roles to the AUR but are handled by the default package manager.
Using binary packages — those that are offered — is missing out on a key strength of Gentoo and the primary reason one may choose it over another.
That all being said, I can answer at least one of these more specifically:
Do you lose any potential control over the system when using the binaries, rather than compiling from source, and, if so, what?
Well, obviously. You lose all of the ability to choose how the software is compiled, which is most of the point of Gentoo: tweaking compiler settings and optimizing for your specific hardware. Binary packages are built with a predefined set of USE flags and are meant to be run on a wide variety of systems.
There also aren’t a ton of them, so you’ll still likely be compiling the majority of your system from source which… may not be to your taste.
TL;DR: Gentoo offers a lot of the same sort of features that Arch users love — AUR vs GURU as an example — and many people use Gentoo with very little tweaking, but binary packages do not fundamentally change much for someone who has never used Gentoo. There will still be a curve and you will be learning.