There seems to be some confusion about the term "firmware", I thought it was the stuff that stays on the pi, rather than the sd-card, but it looks like it may be just "drivers" for the kernel in raspian. Or perhaps rpi-update calls the eeprom update as well?
rpi-update definitely doesn't update the eeprom. It does deliver "firmware" files, I just don't think they are persisted to chip based on what I have seen. I would expect the eeprom has a stage 1 bootloader, then there is a stage 2 bootloader on the FAT partition (which is updated by rpi-update), and stage 2 should load the kernel image. I believe the other bits like "start.elf" may be for the GPU, so those might be firmware files loaded in stage 1 or 2 bootloaders -- but I would maybe consider that a soft-firmware like your Broadcom drivers.
LE is using the same bootloaders, firmware files, etc that Raspberry Pi OS uses -- but the update method & timeline may be different than Raspberry Pi OS, so they could be different versions. And I think the biggest difference is as I mentioned earlier, LE doesn't automatically update the eeprom but Raspberry Pi OS does at boot up. See below for output of one of the systemd services running there, it's dependent on new eeprom files delivered via the "rpi-eeprom" package (once you run apt-get upgrade or via the GUI, install new packages, and reboot it will stage the eeprom for loading, might require 2 reboots, never really looked closely).
In any case, just wanted to mention that it was perhaps the eeprom that need to be updated, in case some one else wanted to use your solution. Good to hear that your solution works on the RPi4.
pi@raspberrypi:~ $ systemctl status rpi-eeprom-update |cat
● rpi-eeprom-update.service - Check for Raspberry Pi EEPROM updates
Loaded: loaded (/lib/systemd/system/rpi-eeprom-update.service; enabled; vendor preset: enabled)
Active: active (exited) since Fri 2022-01-21 21:42:57 EST; 18h ago
Process: 4759 ExecStart=/usr/bin/rpi-eeprom-update -s -a (code=exited, status=0/SUCCESS)
Main PID: 4759 (code=exited, status=0/SUCCESS)
CPU: 138ms
Jan 21 21:42:57 raspberrypi rpi-eeprom-update[4759]: BOOTLOADER: up to date
Jan 21 21:42:57 raspberrypi rpi-eeprom-update[4759]: CURRENT: Thu 02 Dec 2021 11:08:03 AM UTC (1638443283)
Jan 21 21:42:57 raspberrypi rpi-eeprom-update[4759]: LATEST: Thu 02 Dec 2021 11:08:03 AM UTC (1638443283)
Jan 21 21:42:57 raspberrypi rpi-eeprom-update[4759]: RELEASE: stable (/lib/firmware/raspberrypi/bootloader/stable)
Jan 21 21:42:57 raspberrypi rpi-eeprom-update[4759]: Use raspi-config to change the release.
Jan 21 21:42:57 raspberrypi rpi-eeprom-update[4759]: VL805_FW: Using bootloader EEPROM
Jan 21 21:42:57 raspberrypi rpi-eeprom-update[4759]: VL805: up to date
Jan 21 21:42:57 raspberrypi rpi-eeprom-update[4759]: CURRENT: 000138a1
Jan 21 21:42:57 raspberrypi rpi-eeprom-update[4759]: LATEST: 000138a1
Jan 21 21:42:57 raspberrypi systemd[1]: Finished Check for Raspberry Pi EEPROM updates.
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