Is it correct that once enough people get vaccinated, even if we don't eradicate the virus, it will have to mutate into something milder in order to still effectively spread?
It is not a given that it will be milder, just estimated to be more likely once everything is taken into account.
As an example of why, the comment you are responding to mentioned the spike being functional (it is used by the virus for cell entry) and it is also targeted by our antibodies. So, for the virus to evade antibodies, it would have to change the spike enough so that antibodies don't detect it anymore. But, by changing it would likely lose some of its current efficacy.
There's some nuance to this. The current most common strain is theoretically the most genetically fit strain. Wide use of vaccines targeting conserved regions of the spike protein almost certainly would result in less genetically fit virus progeny becoming more common by simple selection. But, there's no predicting the virulence of the the less genetically fit strains that will pop up. ie) very deadly viruses aren't necessarily very genetically fit. The short of it is, the resultant strains will probably be less transmissible (the "R" number will decrease), but it's hard to predict their virulence/serverity.
I think consensus is we don't know for sure, but there's reason to be hopeful. SARS-CoV-2 probably won't evolve as fast as the flu, which undergoes a "sexual-like" evolution process called "re-assortment." On the other hand, CoV-2 has "re-combination," which gives similar results, and it does have zoonotic hosts.
Note there is some correlation between virulence and severity. For something with a longer incubation time, increased viral load tends to mean both increased virulence and transmissibility. This relationship is far from universal, though.
As an example of why, the comment you are responding to mentioned the spike being functional (it is used by the virus for cell entry) and it is also targeted by our antibodies. So, for the virus to evade antibodies, it would have to change the spike enough so that antibodies don't detect it anymore. But, by changing it would likely lose some of its current efficacy.