Can someone create a system that stores and parses all this company registration data to instantly dereference the person behind any shell corporation?

Not just “shell” companies. Berkshire Hathaway is hardly a shell company, but much of its money comes from investments. Same with most insurers.

Maybe shell companies should not exist. Or be fully transparent.

Shell companies, tax havens, tax loopholes... every law has a loophole and guess who makes the laws