Is my bitcoin transaction tracable because of ETH?

[Bitcoin]

Is my bitcoin transaction tracable because of ETH?

I have exchanged bitcoins to ETH through letsexchange.io as I needed ETH. For that, I have used Bitcoins from a wallet which I actually wanted to stay anonymous.
I have used the ETH only for paying fees while swapping my USDC to Bitcoins on another wallet, on Blockchain.
Finally I have send out that bitcoins to the wallet on LocalBitCoins.

My question is:
Can the first Bitcoins transaction or the ETH transaction be traced down and connected to my last bitcoin transaction from the wallet which I only did the swap with the ETH?

I don't want that the publicity can see that the first bitcoin transaction is related to the last bitcoin transaction (like seeing the steps I have done like mentioned above.)
Also I do not want, that authorities could see that.

Source: Is my bitcoin transaction tracable because of ETH?