"Carbon" rims are in no way different than "alloy" rims except in material.
CF rims come in a wide variety of sizes and shapes as do alloy rims.
If you don't need very deep aero rims, then you can get alloy rims which weight less than deep CF rims .... and the biggest difference if "feel" (IMO) comes from reduced rotating mass. The difference disappears once you get up near cruising speed, but accelerating from a stop or low speeds, based on my own tests, the lighter wheels feel faster.
If you generally ride over 18 or so mph, you might gain a tiny a percentage of speed with a set of deep areo rims ... I doubt Strava would notice, and if it doesn't show up on Strava, does it even happen?
When I bought my Fuji, it felt sluggish and I was disappointed with it for a while ... then I got some lighter alloy rims and some Contis to replace the stock boat-anchor wheels and treaded on/off-road tires, saving about half a pound ... then I got some deeper CF rims which were a quarter-pound lighter still. The difference was dramatic from the stock wheel and tires, to CF wheels and Contis, was tremendous ....but the difference was not due to "fast" tires of CF rims .... it was purely Weight. I dropped maybe two pounds and that made the bike feel ... two pounds lighter.