I don't think "higher morals" is a realistic solution within our current market economy - these are multi-billion dollar corporations which operate on a separate set of motivations from any of their employees.
I also don't think our only alternatives are "trust these companies to self-monitor", or blockchain.
I'm thinking about 2 kinds of solutions: 1) technical solutions which would improve the current chain-of-custody tracking (this could involve blockchain, but it doesn't have to). 2) policy solutions which might involve something like "less frequent, but more rigorous testing, with stronger penalties".
And I'm confused by your statement "Requiring submission of soil samples is rather expensive, and fairly strong control, because cheating is a very deliberate act." - but what these instances prove is that corporations will cheat anyway.