I am leaving the depression/inflammation link aside, because I am completely unqualified in that regard.

In COVID-19 there have been some off-label uses of Relafen (nabumetone) in myalgic encephalomyelitis, which is close enough to have some idea. I guess we'll have some publications on that one in the coming months.

For the time being, a review of the literature in https://pubmed.ncbi.nlm.nih.gov/21504126/ seems to suggest that NSAID and COX-2i in neuroinflammation in Alzheimers does not show any efficacy.

https://www.sciencedirect.com/science/article/abs/pii/S0197458099000287?via%3Dihub suggests, that NSAID use may have some efficacy if implemented before inflammation develops. As a purely neuroprotective agent, if you will.

In SARS-CoV-2 infections, use of NSAID may thus, and this is a hypothesis, prevent neurological sequelae. I would caution against hedging a bet on it, though, the data are thin on that one.