I don't think that's true, fees are not proportional to that. Fees compete for a finite and precise limited space in blocks. There's no limit to how high fees could grow in this competition, because only the 7 most lucrative transactions get mined per second.

I think it is vaguely accurate to say that fees and mining costs are linked, *however*, currently the coinbase block reward is a bigger deal. Example: most recent block https://www.blockchain.com/btc/block/743055 created 6.25 bitcoin out of thin air, plus 0.186 bitcoin from all its fees. In the future, when fees make up a larger share of this, miners will indeed start to get income from fees. Then, we will see an interesting dynamic where automatic difficulty adjustments and competition between miners entering and exiting the market will result in miners electricity costs aligning with bitcoin transaction fees. In other words, every unit of value that goes into a bitcoin transaction will result in that much value being spent by a miner on their electricity bill.