Since they been to sea for 200 days, the ONLY way to replenish is UNREPs. If they were hitting ports, they do crane them over. OR they ca anchor out and have stores craned over. You'd be surprised how many ports can let a carrier get dockside (30 to 40 in the world). The bigger reason to not do it is safety / security of the ship in that region and it takes a carrier group "offline" for week or 2+.
You are correct tho, supply ships are the best and pretty fast way, plenty of fresh food can sit for days from the loading to the UNREP event and then stay fresh. It sounds like they've crapped the bed on logistics to keep them resupplied (and as discussed, something weird is going on with the plumbing). Feels like with all the firings, and the unexpected military engagement, the military management is struggling to keep up with the chaos.
Im sure once he is gone, there will be reports and books written about how his methodology left the military off kilter.