depends on distance , how much you need to carry and whether a bike shop is open during your commute times, i work shifts so i have to be self sufficient
a commuting bike in my eyes needs to have the option unless some are using c2w to buy a bike for leisure which is of course a different thing altogether
It also depends a lot on the individual.
I also thought panniers were essential;, but had never tried without. Then the rack broke on my Boardman hybrid (leaving two of the bolts sheared off in the mounting points), just as I was starting to look for a new bike. So I tried with a rucksack, and found it fine. And going without panniers gave me a much wider choice of bike. I do still feel mudguards are essential.
The OP has said the want a carbon frame, and
very few road bikes with carbon frame have the capability to take panniers.
I've been commuting regularly now for over a year (normally twice a week, the other three days WFH) on my Cube carbon-framed bike, 15 miles each way, with my clothes, towel, laptop and lunch in the rucksack (multitool, inner tubes & puncture repair kit are always in the saddle bag, regardless of commute or leisure ride).
I do keep a spare pair of shoes, shampoo/shower gel and spare keyboard and mouse in the office, so I don't have to carry them.
That bike, BTW came in well under the Op's budget, at £2100 for full carbon frame with full 105 groupset including hydraulic disc brakes. But the current version appears to be out of stock everywhere until 2023.