I have an E3372h-320 with firmware version 10.0.3.1(H192SP2C983) (the only difference to yours is a 2 instead of a 1 on the second set of digits) and it's working fine on my Belkin RT1800 running 23.05.0-rc3. I've followed your same instructions, only installling kmod-usb-net-cdc-ether and usb-modeswitch, which keeps the dongle in Hilink mode and presents it to the router as an ethernet device.
There is also a better method, talked about on the thread linked by @AndrewZ, which presents the dongle as a modem. I tried it on my test router Fritzbox 7360 v1 running 22.03 and it worked, the only problem was that IPv6 wouldn't work with it, so I haven't used it on my main router yet (that's probably fixed on 23.05, I'm waiting for it to be available for the 7360 to test). Below are the notes I took when I tested it, try it and see how you get on, if you need help ask.
To use the huawei E3372h-320 in stick mode first follow https://openwrt.org/docs/guide-user/network/wan/wwan/ethernetoverusb_ncm then Huawei E3372h-320 in NCM mode - #12 by woec but you also need to remove all AT commands except "AT", "ATE1" and "AT+CGDCONT" else the interface won't come up. Also note that the msg number is the line number of the messages list starting from 0, best to add it at the bottom.