"...Would a medal engineering company not be a better choice to provide steel mesh tires as Goodyear, a Rubber Company?..."
IIRC tires have steel belts.
...
Steel belted radials were just coming to reality in the years of the various Apollo missions.
No, the 1970 model Lincoln Continental Mark III was the first American-made vehicle with radial tires as standard equipment.
The Citroën 2CV came from factory with radial tires fitted in 1948 as standard.
A steel belted radial tire is nothing like a steel mesh tire where the mesh compresses to the inner-frame under load.
Four major components comprised the LRV tire's design: mesh, tread, inner-frame and hub.

Is like equipping the billy cart with power steering (60lb/27kg)
The approach worked because each tire was only required to support about 60 pounds of weight (on the Moon, the equivalent of 360 pounds on Earth) and be used for a maximum of 75 miles.
Prior to the LRV the MET (Modularized Equipment Transporter) Apollo 14 did have inflated tires.
The MET was a two-wheeled vehicle with a tubular structure 86 inches long, 39 inches wide and 32 inches high when deployed ready to use on the lunar surface. The MET had a single handle for towing and has two legs to provide four-point stability at rest.
The MET was stowed during flight in the Modularized Equipment Storage Assembly (MESA) in the LM descent stage, and was used during both EVAs. Equipment was mounted on the MET for the geology traverse included the lunar hand tool carrier and the geology tools it carried, the closeup stereo camera, two 70 mm Hasselblad cameras, a 16 mm data acquisition camera, film magazines, a dispenser for sample bags, a trenching tool, work table, sample weigh bags and the Lunar Portable Magnetometer.
The MET tires were 4 inches wide and 16 inches in diameter, and were inflated with 1.5 psi nitrogen preflight. The tires were baked at 250 degrees F for 24 hours preflight to remove most of the antioxidants in the rubber. Operating limits for the MET tires are -70 deg. F to +250 deg. F.
Empty weight of the MET was 26 pounds, and the vehicle had a useful payload of about 140 pounds (Earth weight) including the lunar soil samples to be brought back to the LM from the geology traverse.
Estimated travel rate of a crewman towing the MET, as determined by tests with the 1/6-g centrifuge rig at MSC, was about 3.5 feet per second, with a one pound of pull required on level sand.
From Press Kit, Release No: 71-3K, Project: Apollo 14.

Judging by the footprints the cart is still at base and not attached to power source.