Thanks for the help @hecatae. It worked in the end!
I continued using firmware 1.0.0, set tp_boot_idx to 0 with fw_setenv, but after rebooting the router, the firmware went back to version 1.0.2 Build 20230808 rel.59994(4555). Once this happened, I went ahead with the installation steps from the start and got OpenWrt running.
To summarize what should work for someone starting from scratch:
a) If you've got v1.2 of the MR90X router, continue with the pre-installed version 1.0.2 firmware. You can confirm the version from the router's web interface, at the bottom of the login page.
b) Gain root access via Telnet as described in the OpenWrt Installation instructions. Run fw_setenv tp_boot_idx 0.
c) Reboot. You may do it by running reboot from the Telnet session.
d) Perform all the steps in the OpwenWrt Installation instructions from the start. Ignore comments about tp_boot_idx in step 3 and proceed with subsequent steps.
e) You should be able to log in via SSH to 192.168.1.1 on port 22 with root user (ssh root@192.168.1.1).
For definitive instructions, one would need to confirm whether steps (b) and (c) are really necessary. It's possible that following all standard installation steps would work directly.