You're trying to imagine Game Theory will protect bitcoin ,
and it would if bitcoin was the only network the miners could mine.
But since the Miners have access to mining multiple networks, they will be true to their greedy nature and play them against each other,
untiil they decide for certain which one gives them personally the most profit.
Which due to the decreased rewards and ever growing reliance on transactions fees,
the transaction fees profit will lead the miners away from bitcoin as LN will dominate it's fees.
There's very little to no competition for Bitcoin's hashpower which probably won't change for two reasons:
1) Transaction-fees-only from a highly valued cryptocurrency is still more profitable (ie. able to pay for more hashpower) than transaction-fees-with-block-subsidy from a cryptocurrency that has very little value. LN is unlikely to "steal" fees from miners as there's still transactions to be made and mining fees to be paid -- just shared across a much larger user base, decreasing the cost for the individual. That's if LN succeeds. If it fails its impact on mining fees will be -- obviously -- nil.
2) The largest cryptocurrencies sharing their hashing algorithms with Bitcoin are (currently) Bitcoin hard forks that follow the same emission rate, so they have no advantage in terms of transaction-fees-with-block-subsidy vs transaction-fees-only. All other SHA256 coins are way out in the water, the introduction of new SHA256 coins (apart from the stray Bitcoin hard fork) is unlikely to happen.
That being said, what's your opinion on Ethereum's Casper?