Broadly speaking, this is opposed in Canada by some due to the perception that the existence of a private parallel system will be weaponized to undermine the public system, as many feel has happened to the NHS in the UK.
Public Healthcare absorves the heaviest blows: expensive surgeries, cancer treatment etc are mostly or only done by the single payer public system, while private Healthcare enjoys mostly low severity cases.
If there was no private system, the public system would pay for all of that. What you’ve described actually seems like it goes easier on the public system by allowing it to specialize on certain kinds of issues
The basics are actually the thing it does best; covid vaccination for example went well once the political stuff was handled.
It seems like the ceiling here is the amount of investment it would put in to manage 100% of the health needs —- I don’t see how the parallel existence of a private system causes the public system to be larger than it would be if there were no private system