I guess it depends how you view your obligations. If you don't want people to be able to read, vet, or reproduce your research because the citations are suspiciously tricky to follow up on, that is of course your prerogative. As a scientist free and fair access to accurate and easily verifiable information should, of course, be in the forefront of what you care about. It is completely doable as well -- I've done it with my research for years. Even when citing books, I always ensure the relevant passages are available in Google Book search or something like SciHub. If not, I won't cite that source, because I wouldn't have been able to access it in the first place anyway.
When we cite a paywalled paper, we muddy the integrity of our field by saying "sure, you can check up on this, if you pay that is". The downstream effects of everyone doing this is that there are quite simply more inaccuracies out there in current publications. Replication studies literally become more expensive because to replicate 1000 papers, you have to get by 1000 * x paywalls, so we get less replication studies, so the field suffers. If we want accurate research, everyone needs to be able to access every single citation without issue or highway robbery just in case a janitor without JSTOR access finds a flaw in one of your citations' differential equations.
This also leads to the (damaging) centralization of power -- only large institutions can afford to do things like replication studies, and only academics at large institutions can afford to read paywalled citations. It's like we're living in the dark ages. This made more sense when paper and printing were incredibly expensive, like 300 years ago.
In this way, institutions paying for these subscription memberships actually does damage, since they are keeping this model profitable. They are subsidizing the destruction of integrity in fields they care about.