It's not unreasonable to argue for socialized healthcare based on treatment denials in private healthcare, since there are impactful differences in the incentives driving denials and rationing in private vs socialized healthcare. I agree that the argument should be more nuanced than just "denials happen".

The incentive for private health insurers is to raise prices and increase denial rate until people are unwilling or unable to pay. People will pay until they can't, since they don't want to die, so this can be pushed pretty far. The incentive for socialized healthcare, at least in principle, is to provide people with as much treatment as is feasible for the amount of incoming funds. In one case rationing is driven by a need to remain solvent and in the other case it's driven by profit maximization. The different incentives lead to significant differences in how people are impacted by the denials/rationing that necessarily exist in both systems.