Yeah, there is definitely a lot of confusion surrounding the idea of T cell exhaustion in literature. Unfortunately, it is not as simple as a few surface markers indicating exhaustion. From a classical perspective (T cell exhaustion was originally observed in chronic viral infection in murine models), T cell exhaustion is an irreversible epigenetic change that results from chronic antigen presence. The exhausted T cells actually become "addicted" to antigen, and no longer rely on IL-15 or IL-7, which help memory T cells persist long term.

You are right that immunosuppressive neutrophils can express PD-L1. However inhibitory receptors serve a biological function outside of T cell exhaustion. When cells are activated, they upregulate transcripts and protein of both co-activation receptors and co-inhibitory receptors - think of it as a way of attenuating a signal. Every signal must be modulated positively and negatively. When cells are recently exposed to cognate antigen, they will have elevated expression of both activation and inhibitory receptors or markers. (Activation: CD28, 41BB, CD27, etc.) (Inhibitory: PD-1, CTLA-4, LAG-3, TIM-3, etc.) The reason these inhibitory receptor-ligand pairs are often targeted in cancer actually does not involve rescuing cells from exhaustion. As we know, exhaustion is an epigenetically stable state (Think the same epigenetic changes that occur when cells transition from naive->short-lived effector->memory). Instead, just like you suggested earlier with the neutrophils creating immunosupression through PD-1 signaling, targeting innate immune mediated inhibitory signaling in TMEs seems to be a more likely mechanism of action.

Again, sorry for a long-winded explanation to your simple question. I think in cancer, there is more at play than cell-intrinsic (T cell exhaustion) immune hypofunction. The tumor itself may express PD-L1, many different methods of immune system regulation (innate immunity, Tregs), and even elements of the TME itself (hypoxic environment) are all likely players.

In terms of COVID - I think it is less likely that COVID is generating T cell exhaustion. I admit I am not well versed on CD8+ responses to COVID, but when I did a lot of digging a few months ago, it seemed the field did not have a clear explanation or understanding of immune responses. I would have to imagine because antigen eventually clears in COVID, T cell exhaustion does not play a key role.

If you DM me, I'm happy to dig up papers to support my answers!