It would be intellectually dishonest, if the foundations of your reasoning were sound.

The increase in the money supply, aka block subsidy, is a value transfer from BTC holders to miners compensating them for enforcing the consensus rules of the ledger. Very similar to if you were paying fees to a mega secure vault company, though no vault company can provide decentralised value storage; value storage without any single point of failure. Considering the unprecedented level of security on offer, the value for money is astonishing.

The miners job is to construct a block.
They're paid via a special transaction called a coinbase (a large American exchange named their company after this special transaction), which they can make payable to whomever they wish.
They take incoming transactions and may include a certain number of those; the limit is 4 million weight units, as opposed to a certain number of kB as is commonly believed.
Each transaction they include will usually include in addition to fixed recipients, a special "spend to anyone" transaction output, this is to encourage a miner to include your transaction preferentially as they can make this portion payable to themselves.
They may attempt to include multiple transactions spending the same funds to different recipients, or to insert a coinbase transaction wherein they have made payable to themselves any number of bitcoin, or include more than 4 million weight units worth of transactions, or make some other attempt to break the rules of consensus that any block they published will be tested against by thousands of other computers who will then choose to store locally the new block as the new strongest valid chain tip, or disregard.
Before publishing the transaction though, the miner must also solve the double spending problem, by providing proof of work.
They do this by going back to that block they're constructing, and firstly putting a random number in the nonce field of the header. Then, they pass the blocks bits into a sha256 hash function, take the resulting hash and hash it again with the sha256 function.
The result must meet the current block height difficulty, for example to just happen to start with 18 leading zeroes.
If it doesn't, the nonce is incremented in the header and the block is hashed again.
Without an appropriately difficult proof, first of all you wouldn't know which valid block had come first, and so would have double spend problems where after receiving payment, the block containing your payment was dropped and replaced with another, or outside of bitcoin, anyone can for example send you a transaction receipt via e-mail stating you have been paid, on its own it's worthless though, you will have to rely on PayPal or a bank to ultimately tell you if you have received a payment, and PayPal or bank payments are reversible in a way Bitcoin transactions are not so even then, you don't have the same degree of certainty (furthermore your funds are not in your custody, instead you have lent your money to the bank; they owe you it).
I hope this clears up the difference between these different fees and what they are in payment for, please DYOR next time before speaking on a subject as though you have some knowledge of it, much of the confusion around Bitcoin results from people who have not studied it wanting to sound knowledgeable.