Well, it would have been pretty exactly this had you decided to install the steam snap that is well integrated with Ubuntu, bundles all the i386 bits it needs, has been developed by a team of Canonical employees and gets regular testing…
You picked the path of using a third party installer instead which is simply not as integrated…
While the steam developers could have added a hack to their package to automatically enable the i386 arch (by breaking Debian packaging policy) they decided not to even though you gave them full root access to your machine