oseltamivir is an analog of N-acetylneuraminic acid (like shikimic acid in star anise)...there is a protein on influenza called neuraminidase and in coronaviruses, hemagglutinin esterase, that recognize these glycans on host cells. Oseltamivir inhibits these viral surface proteins after binding to them.

I am not sure how it affects the immune system though.

Please see the following RG links.

Article The administration of oseltamivir results in reduced effecto...

Article Immunological effects of the orally administered neuraminida...

To date, I failed to find out an authentic report depicting the correlation between the consumption of oseltamivir(Tamiflu) and the incidence or mortality of COVID -19 patients.

I just recently found that SARS-CoV-2 (and SARS and MERS) do not have hemagglutin esterase fusion (HEF) proteins because they are lineage B betacoronoviruses (they have the spike that's responsible for infection after binding to their target receptors).

It's lineage A coronaviruses that have HEF proteins (function is essentially that of both hemagglutinin and neuraminidase), since they do not have a spike protein to infect the cells.

Because of this, oseltamivir may work on other coronaviruses, but not for covid-19. Or at least if it does inhibit SARS-CoV-2, it would likely be through another mechanism.