macOS arm64 cannot find x86_64 versions of applications

My macOS 14.2.1 (Darwin 23.2.0) is arm64 by default. I set up a bunch of x86_64 versions for different apps (have Rosetta 2 set up as well). For example, cmake and Homebrew. But when I call them with arch -x86_64 prefix, the OS cannot find them:

arch -x86_64 cmake
arch: posix_spawnp: cmake: Bad CPU type in executable

The following thing works though:

arch -x86_64 /usr/local/Homebrew/Cellar/cmake/3.31.4/bin/cmake

The same is with the other apps installed for x86_64. Even when I call arch -x86_64 zsh and switch my bash to the x86_64 mode, it can't find the paths. When I call

which cmake

It always shows only one path (arm64), either when I'm on arm64 or x86_64 version of zsh:

/opt/homebrew/bin/cmake

Should I manually add all paths to the x86_64 applications to $PATH or is there a better way to tell the OS to do this automatically?