The engines they use in the game (yamaha) do not cost $10000.
With kart racing in oz they implemented wet tyres for racing a few years back. Before that it was slicks in the rain and when it rained you put a plastic cover over the air box to stop water getting in it.
Even if it rained while you were out there, you just turn your air box to the side to stop water getting in. If it does get in, I never saw one engine get destroyed. The water pools at the bottom of the air box as it is on a angle, but even that water was very little.
The Karts in the game in general do not represent top tier karting. The karts in the game are straight chassis, single gear, locked ratio karts that more approximate somewhere between level KF1 and the lower KF3 (I'll use internationally recognized abbreviations from here to keep consistent).
In most series with that spec, beyond endurance karting and special events, rain tires are not used outside of specific kart championship races.
You're right about the airbox, and the engines, ive not seen an engine destroyed by rain either. However the main two reasons are safety, and the second reason is cost effectiveness.
I've fielded two karts over the past five years (what are essentially KZ1 spec shifter and KF1 spec I use mainly for enduros) if I had to spend money on replacement rain tires for my KZ1, I would never compete. The KF1 can use the same tires, but only for enduros.
Most tracks simply shut down in the rain for safety reasons here. Like I said, EU and others might do it different.
$10,000, that's insane. No wonder so few people can do karting when costs can reach that amount!
I see what you mean about braking, first corner at eiger in the wet the kart will always lock up so you have to brake for almost 3 times as much in order to slow it down for the corner.
In the top tier of karting (which im not involved in) it can cost as much to field a superkart as it costs to field a formula ford. However, 10k per engine is a bit unrealistic unless your in the world championships. Depending on the series, a decent kart spec (Shifter or otherwise) will run about five to seven grand per turnkey kart.
Tires are were the main costs lie, and engine if maintained will last alot of races provided your driver isn't stupid and you set the gearing properly to not bang the hell out of the limiter.
Back in the 90s, I used to pay 2$ a lap at local kart track that ran offset "Karnykarts" which were basically repurposed oval karts. I used to go down on rainy days because I thought it was fun to run in the rain (back when they couldn't just shut down the track and refund money and remote-kill was something of the future) I've known people to get injured doing that, and stopped myself once I matured. Never seriously mind you, but I guess safety > racing nowadays.