Incorrect.
"The miner subsidy" is actually composed of two parts:
  1. The block reward, which is cut in half every 4 years, and its newly created BTC.
  2. The transaction fees from the transactions the miner includes in the block.
The block reward will eventually reach zero.
The fees go on forever.
You're worried about a flawed system that exists only in your mind. Bitcoin is perfect.
The transaction fees alone will be sufficient to secure the network against all attacks in the future forever.
Firstly, transaction fees are not part of the subsidy, they are part of the block reward. You have your terms mixed up.
The subsidy is initial distribution of Bitcoin. Inflationary monetary policy The fees are payment for services
Both make up the block reward
Secondly, I am not that worried. Peter Todd who does Bitcoin core testing and things, is worried and you can read that here:
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