Starting the init system is the task of the root filesystem or initrd, with any boot loader. Systemd-boot happily boot into any init system just fine, just like any other bootloader that can boot Linux will boot into systemd just fine.
Systemd-boot boots kernel images (with efi-loader code embedded) and only offers a menu to pick which kernel file to load. What makes systemd-boot interesting is that it does nothing more than that: It does not read random filesystems, it does not implement random encryption things, does not parse image files and complex theme configuration, ... .
An update is only truly done once no remnants of the old code is in memory. Code can stick around in the form of binaries (restart the binaries), libraries (restart all binaries that use this library) and the kernel (reboot or use kexec).
One very simple way to make sure no old code sticks around is to reboot:-)