There's nothing to do with the exchange volumes but just as said by xalolex, it's about the erc20 tokens demand. Well, we can say that there's something to do with the exchanges because it's being traded there and most people are sending their erc20 tokens into exchanges to trade it.
But in actuality, it's just all about the production of many erc20 tokens and the demand that they're having which is making the network clogged which is turning the fee to be higher than before.
The ERC20 token demand is certainly the "main" reason why it keeps going up, or when the times comes it goes down. However, that doesn't mean that it is the only reason, of course exchanges and many other things causes it to go up or down as well, this is a multi-function coin and there are many ways that you could use it.
You could use it to pay for gas fee on your ERC20 tokens, you could use it to stake, you could use it to just buy and hold, you could trade it, you could buy NFT with it and many other things.
This means that there are many reasons but you are 100% right that the biggest reason is that people use ETH for paying gas fee, and during bull runs like we are having right now, the gas fee goes up; which means there are more transactions are happening on ethereum network for some reasons.
Those reasons has been said and you've said most of them. The network going with some traffic really makes the gas fee high. But let's also recognize that the fees have gone down.
Before the highest was like 200 gwei and now, as of typing this, the gwei is just around 22. And that's a good progress IMO and we're so close to the going down of the fee.