I don't think it's necessarily a function of fiat price of bitcoin, though it is correlated. The popularity of bitcoin, or more specifically the velocity of transactions entering the mempool is what matters here.

Here is how I see it. Bitcoin transactions (and fees) are, in its simplest form, an auction system. Some people need to get into the next block and will be willing to bid more. These are the transactions that will ultimately drive the cost to transact (fees).

Yes, some people can wait and will bid less but they will have a lesser impact on pushing the price of blockspace since their transactuons will just sit in the mempool until average fees drop to a point where they finally reach the minimum price to be included.

These lowball transactions don't necessarily pull down the median fee to transact since they just sit in the mempool. No one is necessarily competing against them racing to a price floor. This only happens if we have partially full blocks and an empty mempool. If that ocurrs, lowball fees will drive fee price. I doubt that will ever happen.

While the large overpaid fee transactions do increase the median price and over time, continue to pull up the average price to transact. As such, when looking at median fee to transact over time, due to the nature of the acution system and blockspace scarcity there is a natural pull for fees to increase, not to decrease.

The point here is there is a median fee paid which establishes a general baseline cost to transact. This baseline cost has no bearing on the size of your UTXO. Though the size of your UTXO eventually will matter.

As more people use bitcoin (or more precisely as more transactions enter the mempool bidding for limited block space) the median price/vb will naturally go up as more people bid against one another for blockspace. It's a simple function of scarcity/price as I outlined above.

Eventually, if the average median fee over time continues to rise based off the concept of blockspace scarcity, then your small UTXO will evenatually get swallowed up by the cost/vb to transact. As it may end up costing as much in fees to transact your small UTXO as there are bitcoin represented by that particular UTXO.

The fiat price of bitcoin is simply a loose correlation with the "popularity" of bitcoin. As the price goes up more people get interested and try to enter the space and make a transaction.

Ultimately I think the fee/vb is driven mostly by the velocity of transactions entering the mempool. Or more specifically the velocity of new transactions entering the mempool that exceed the minimum fee required to have been included in the previous block. This may be a whole other discussion but I say it's based off the previous block because no one really knows what the lowest fee for the next block will ultimately be until it is mined.