The line of reasoning isn’t wrong, just incomplete imo - but mostly right.
A retailer has to pay all costs. One of the sources of revenue with which costs may be paid is sales revenue [prices] - indeed sales revenue is the only indefinite revenue stream for a retailer.
However if one particular retailer [in a competitive environment] is unable to recover increased costs via sales revenue, it only means that the costs must be recovered from somewhere else.
In the end, it is always an individual who pays for [increased] costs incurred by a retailer - always.
It is usually an individual consumer via higher prices. But if the retailer is put at a competitive price disadvantage for some reason [as in this case] then the increase will have to be made up by an individual elsewhere;
it will be paid by an individual worker in the form of lower wages or
it will be paid by an individual investor in the form of lower ROI.
Those [or a combination defined by market circumstances] are the only three possibilities. It sounds like CVS determined that neither wages or ROI can suffer. Hence, no increase in cost will be permitted.
Bingo, some firms will choose to eat the increased costs because they don’t want to affect their demand curve. To say that it will “always” be passed to the consumer via higher prices is wrong.