I have had two Elite Accounts since 2016 and had no trouble switching between them.
One is out in the black and one in the bubble, also both have been updated to Odyssey.
I haven’t played for about 6 months and recently logged on to check on my carriers.
I never had any trouble with Steam only having one Steam account.
On the upper part of the Elite Dangerous loading screen is the log out button that for me just launches the game…… strange
I may have to write up a bug report, but there bug system is a real pain and to be honest mostly useless.
However you say that you had the issue when changing accounts when using Steam. The new launcher uses "Log In With Steam", in which case case any attempt at changing your commander account on the launcher will not work, having logged in through Steam it presumably stays logged in to that Steam-linked account.
All you have to do is launch the game outside Steam (since your accounts are pre-2019 you can even run a Steam-bought version without going through Steam).
Whether you make a separate non-Steam installation or try to run EDLaunch.exe in the Steam installation file structure (without having Steam running) is up to you. I run a separate installation for my non-Steam commanders but have one account that needs to be run from Steam so I have Elite installed there too.
TL/DR - don't run from Steam.