Why does the "global lottery" prove the validity of the blockchain? As far as I understand it currently, when winning the "lottery" (mining) you can add a block, sum of transactions, to the blockchain. This block is in the first instance verified by all network users.

BUT can`t a lottery winner also add a fraudulent block to the chain?

What does this lottery contribute to the validity of the chain?

Or in other words: How are miners approving a bitcoin transaction, besides adding it to the blockchain when they win? Potentially, they could also add a lot of "sh**" to the blockchain.
Miners do not prove validity. They solve the problem of needing to decide between two potentially conflicting but otherwise valid versions of history.

The rule that full nodes follows is that they accept the longest valid block chain, where valid is defined by the network's consensus rules, and long is defined as having more mining work performed. If a miner produces a block that is invalid according to those rules (for example, it spends a coin without valid signature), full nodes will ignore the block and any block built on top.

There are also lightweight nodes ("SPV nodes") which do not verify validity at all, and will accept whatever block chain has most work. Their security however relies on the fact that blocks are expensive to produce, so a miner wouldn't create one that isn't accepted by full nodes.