Why are airlines so concerned with checked baggage weight?
Airlines do care about your carry-on weight and even your weight. Many have carry-on weight limits, which low-cost carriers usually enforce, weighing the carry-on and tagging it as cabin luggage. Full-service airlines rarely bother.

One reason they don't measure your weight is that most people wouldn't feel comfortable getting weighed in front of strangers, then charged for that. Only Air Samoa currently weighs each passenger and tickets them by weight. In the West, weight is a contentious issue, so an assumed average is used for passengers. Such assumptions have crashed small planes, and in special circumstances like sports charters, actual weight is used. Finnair weighed volunteering passengers to get a better sample.

Weight costs money to carry, and not just in fuel burn. Luggage shares its volume with cargo, which airliners from the A320 up are designed to carry in the hold. Lately, cargo tends to get become more profitable than passenger service. Electronics and other high-value items cost more in depreciation and time value to ship by sea and land than by air, creating massive demand for air freight.

The A320 is 42.6 tons empty, but that's not the number that matters. The number that matters is its 78 tons MTOW, and how short of that it comes after adding pax, luggage and fuel to the 42-ton OEW. That spare weight is money in standby cargo. 

Giving potential 175 passengers an extra 20 kg each (long-haul FSA vs average LCC allowance) could, if everyone used it all, add up to 3.5 tons total. Freight rates vary widely, but on a 3,000 mile flight that's worth \$7k-\$10k in freight revenue, or \$40-\$60/pax. Admittedly that's on the upper end of the range, but you can see there's money to be made. For an A320 that only costs ~$7k/hr to fly, it's a lot.

Less ethically, moving weight and volume from luggage to carry-on is profitable for a mixed passenger/cargo airline. Checked-in luggage has to be weighed in all cases, to ensure each piece is within safe limits for manual handling. This known weight then eats into the takeoff weight calculation, leaving less for cargo. 

As airliners are designed to take off with one failed engine and carry reserve fuel for failures and diversions, this extra power works as weight safety margin when nothing fails. In practice weight margins rarely get pushed - most flights are volume-limited, with enough weight margin to take off at reduced power. Still, less luggage means more cargo means money.