Do gas fees increase when price goes up?

I understand that gas fees go up as network traffic goes up. However, I'm struggling to understand the process behind it. To give an example: Let's say Ethereum is on $4k. Let's then assume that gas fees are approx. 0.01eth so you have to pay $40 to do a swap on uniswap. If Ethereum's price were to reach $8k, wouldn't that 0.01eth would now cost $80? If price goes up, doesn't the gwei need to go down, and if so how does that happen?

Rn it costs around $100 for me to swap then I need to sell (so $200 all together). I'm just worried that if hypothetically an ETF comes out and ETH moons to $20k by 2023, we'll need to be paying $1k to swap and sell. Surely that can't be the case. I know rollups and sharding will help bring these prices down, as will increasing the block space, but is the title of this post incorrect?

Thanks!!