AFCI breakers and code compliance
What does the citation mean?

The facts are that you need to apply the new codes inside the scope of new work, and the remodel qualifies as new work.  The areas of the house you are not touching should be grandfathered.  The section of the code you are citing did not mention that fact.  This is a common problem with citing random bits of code.  

The code says you need AFCI protection in all those places.  However this is overridden by the fact that the other locations are grandfathered.  The area you are doing new work is not grandfathered and the requirement applies there.

New panel

You can replace your existing panel if you really want to.  Or if it is a dangerous FPE.  Or if it is an obsolete panel like Pushmatic where you won't be able to get AFCI breakers.   

However, you can also install a subpanel.   You would need to move a number of circuits over from your existing panel to the new subpanel - enough to make space for a large double breaker, which will power the subpanel.   

Now if you need 2 extra circuits and need to move 4 to make the space, you can get a 6 space panel.  However that means you'll be right back here again next time you add anything.   You could add a 12 space panel but again you are just kicking the can down the road.  This is where my colleagues recommend a 20 and I recommend a 40 space panel.  The cost is trivial compared to the cost of having this problem.   My first motive is to never have to deal with this again, ever, and my second motivation is to plan ahead so the subpanel can later become the new main panel, after you move circuits over, one at a time at your leisure.  

Also beware of panels which say "50 circuit 30 space", those are only good for 30, because the double-density breakers are not available in the GFCI or AFCI that almost every circuit requires these days. 

The subpanel can be next to the main panel (a good idea for a future switchover).  Or, if the kitchen is fairly far away from the main panel, it can be near (not in) the kitchen.  That will make all your circuit runs much shorter.  

Everything related to the new panel work will lose grandfathering and need to meet current code.   This is one reason to do the subpanel instead of replacing the main panel; it avoids having to upgrade every circuit to new standard, almost all both AFCI and GFCI.  

While we're on the subject of overbuilding...

Also, what sort of activities do you expect in the kitchen? Any chance someone might be, I don't know, just spitballing here... cooking?  Because these days, a lot of kitchens see not much, while others see quite a lot.  Those can be populated with coffeemakers, toasters, skillets, grills, sous vide, wafflers, you name it.  These heat-making appliances are typically sized at 1500W and cannot share a 20A circuit.  And your chef will probably want to use several at once when busy.  S/he will mightily appreciate it if your kitchen is populated with as many kitchen countertop circuits as you can bear to install. One circuit per receptacle is not excessive.   Crazies like me even go for one circuit per socket.  This is where a subpanel near the kitchen will be particularly useful, and that's the easy way to do it.