This sounds like the issue I’ve had. Basically, the Broadcom adapter is supported by a couple of different drivers, most notably ‘b43’ which is a reverse-engineered driver, and ‘wl’ which was written by Broadcom. The ‘wl’ driver is the preferred one in most cases and its installation disables the automatic loading of the other one.
The wl driver is also loaded by the kernel on the installer disk but, personally, I’ve never gotten that one to work correctly. Fortunately, there are a couple of options:
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Install EndeavourOS using the “offline” installer. This installs an XFCE4-based desktop which will include a wl driver that will work on reboot.
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Do the initial install using a wired Ethernet connection to your modem/router.As in the first case, a working driver will be installed.
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If you can grab a copy of /lib/firmware/b43 from another Linux system, copy that directory to a flash drive other than the one with the EOS installer. Boot the installer. Once in the installer Live CD, insert the drive with the b43 firmware into another USB port. In the file manager, open a terminal in that drive and type the following commands:
$ sudo -s
# rmmod wl
# cp -r b43 /lib/firmware
# modprobe b43
At this point, you should be able to connect to the network normally.