The lottery isn't the point, it's a side effect. The network needs consensus on what transactions have happened and which haven't. The main difficulty in getting this when you can't trust the nodes that someone can just spin up an arbitrary number of nodes to overwhelm the consensus, if the consensus just goes by majority vote. If you want to prevent this you need to tie the consensus to something which is scarce, which is what proof of work does. As a side effect of how this is implemented, it becomes a lottery.
If you implement proof of stake, for example (where the scarce resource becomes the currency of the network itself, which is a lot trickier to implement and especially quite hard to bootstrap from nothing because in the early days of a currency there's going to be only a few people owning it), then there's no need for the reward process to be a lottery (though it still can be).
It's still a lottery, except the way you buy tickets is by wasting electricity. The more electricity wasted the more lottery tickets you have. It doesn't need be like this, just buy the tickets with effing bitcoins and you achieve the same result without wasting electricity.
It’s not that easy. You can’t secure Bitcoin with a lottery run on Bitcoin, because then you could forge your own tickets and forge the drawing, too.
The proof of stake proposals don’t seem to work that way. They seem to be more a matter of stakeholders having machines that vote about who is trustworthy.
I don't think you would be able to forge your own ticket. For example, the lottery tickets could be "purchased" by sending some amount of BTC to a special/nonexistent address, and these transactions would be required to be present in the block being validated for the block to be considered valid. Admittedly I haven't thought about all the details but it would surprise me if there wasn't an obvious alternative to the proof-of-waste system currently in place.
If you implement proof of stake, for example (where the scarce resource becomes the currency of the network itself, which is a lot trickier to implement and especially quite hard to bootstrap from nothing because in the early days of a currency there's going to be only a few people owning it), then there's no need for the reward process to be a lottery (though it still can be).