But it would still be more expensive, and still have a latency issue.

Imagine a public counterstrike server where there's an extra 30-50 ms delay between when you hit the strafe key and when you start moving.

Alternatively, Counterstrike or another shooter could defeat wall hacks if the server only told the client about player positions in the client's line of sight. But then the Counterstrike player executing a peek would see their opponent pop in 30-50 ms after they gain line of sight. Much Counterstrike gameplay is built upon the short interval between when you see someone, and when you click on their head with your hit scan weapon.

Furthermore, latency is not going to go away for Internet play. The speed of light travel time to circumnavigate earth is 125 ms. That gives a theoretical worst case minimum ping of 62.5. Actual pings I see from my ISP are approaching the speed of light order of magnitude, assuming they are only traversing North America.