If you assume a 600 lumen output per can, to achieve 35 lumens per sq ft (overall), you need 6 - 7 cans. Besides CFL, LED cans are also available. The Cree LR6 works well and you could consider the cheaper CR6. I used the LR6 as the light was much nicer than cans using CFL. My kitchen is better lit today than before, when I had a 7' ceiling with a recessed light box with t8 lights. Some places (eg - California) require compliance with energy efficiency code (eg - title 24).

You will find more info in the link below. Here is a link that might be useful: Led recessed lighting kitchens & more

We had ceilings that were a little less than 8 ft in our last house and used 6" recessed cans. We also used them throughout this house, which has 9 ft ceilings downstairs, but 8 ft ceilings upstairs. They have the white trim rings, so blend with the ceilings and I think they're fine. I don't know if it helps, but here are photos that show the ones in our bedroom and in the fridge/pantry alcove of our kitchen, which both have ceilings at 8 ft. The one in the kitchen shows up more because of the green ceiling, and the camera captures a glow that makes it look more obvious than it does in person.

I have to agree with your designer friend: I too believe 6" cans are outmoded. I feel even 4" cans look dated, but look a lot better than 6". Our 10' high flat ceilings in the dining and kitchen (30'9" x 19'8") were completely redone (they were tray and vaulted before), and we put 26 3" mini-cans with 3.5W LED bulbs for general lighting, 16 3" mini-cans for task lighting (4 over main sink, 8 over main island, 4 over second island), 10 50W PAR20 incandescent bulbs in our 72" hood being replaced by 5W LED bulbs, and 3 pendants over the dining table which will have CFL bulbs in them. I have replaced eight 6" can lights in my living room with 14 3" mini can lights with 3.5W bright white LED bulbs (350 lumens). While only 10 lumens/sq ft, this seems to offer adequate light for our 25' x 20' living room with 13.5' ceilings, with white ceilings, walls, and white marble (Botticino Fiorito). We have about 60 4" 12V MR16 can lights, and we've replaced the 40W halogen bulbs with 3.5W bright white LED bulbs. We have slightly less general light in these rooms than with the 40W halogen. Your six 4" LED cans will likely be bright enough at 27 lumens/sq ft, but will depend on your wall, floor, and counter colors (darker surfaces absorb light, light surfaces reflect and diffuse light). I have 14.7 lumens/sq ft of general lighting, which feels a bit dim on their own despite white ceilings, white counters (Caesarstone Nougat), and white marble floors (Botticino Fiorito). With the island lights on (13.5 more lumens/sq ft), we have very good, bright light with 28 total lumens/sq ft. With the over-sink lights, range hood lights, pendants, and 41 lineal feet of in-cabinet accent lights on, we will have football field class lighting.