I concur with your hypothesis. After battling similar symptoms for 40 hours straight, I've discovered this confirmation:

• https://packages.ubuntu.com/impish/gcc-10-aarch64-linux-gnu
• https://packages.debian.org/bullseye/gcc-aarch64-linux-gnu

Note that Ubuntu 21.10 (Impish) and Debian 11 (Bullseye) have packages for a gcc 10 cross compiler. Be wary of the very confusing fact the Ubuntu's default package is actually gcc 11, but Debian 11's default is gcc 10. The similar version numbers of Debian and gcc are a coincidence. Also ignore for now the fact that Ubuntu's package is gcc 10.3.0 and Debian's is gcc 10.2.1.

Focus instead on the recommendations and dependencies of each package. Ultimately the Ubuntu package calls up libc >= 2.34, while the Debian package calls up libc >= 2.28.

Sure enough, when I cross-compile from Impish on x86 for Bullseye on aarch64 (despite having a complete SYSROOT for the target), I get this at runtime:

/lib/aarch64-linux-gnu/libc.so.6: version 'GLIBC_2.34' not found

But your question remains, is there any tie between the host libc and that used by the cross-compiler? The answer is a definite maybe.

See this excellent answer and links for an overview of a cross-compiler. The take-away:

You don't just cross-compile glibc, you need to cross-compile an entire toolchain. Toolchain components are ALWAYS: ld + gcc + libc + gdb.

So the C library is an integral part of the cross-compiler.

What shenanigans then, are going on when you install gcc-aarch64-linux-gnu? It's just a compiler - only one of the four parts of a toolchain.

Well apparently there's some flexibility. Technically, a cross-compiler can be naked. That's typically only useful when you're compiling an operating system, rather than an executable that runs on an operating system. So you can construct special toolchains for special purposes.

But for the standard purpose (cross compiling for Linux on another architecture) you want a typical toolchain. Which is where the package's dependencies and recommendations come in. A gcc is always in want of an ld which is always in want of a libc, and the ménage à trois is intimate. In fact, gcc is built with libc using ld in a complex do-si-do. See this example from a great guide by Preshing on Programming:

It's possible to force separation and link to other libraries, but it's not easy.

For example, the linker you use has a set of default search directories that are baked in. From the fine manual:

The default set of paths searched (without being specified with -L) depends on which emulation mode ld is using, and in some cases also on how it was configured.

And it gets more intwined. By default, gcc will call on a dynamic linker whose location is hard-coded. For a cross-compiler, it might be something like /lib/ld-linux-aarch64.so.1. Not only that, the executable may also end up with the hardcoded path, as its program interpreter.

Again, if you're careful you can tear apart the toolchain and override things. But not only is it tricky to enforce, particularly if you have a complex build, the multitude of combinations of options and paths means there are also often bugs. So your host environment can easily leak into your cross-compiling toolchain.

So in summary, cross-compiling requires a toolchain. While pulling a cross-compiler from a package manager seems like an easy and legitimate thing to do, it comes with a lot of implicit baggage. You can either carefully follow the package dependencies to check what version you're getting, or use one of the many dedicated toolchain environments, such as crosstool-NG.