"Today in January 2019, I created a liability check, but dated it October 10, 2016"
Let's review: you are Catching up the bookkeeping. So, you date it the same as if you just did it = Your Date = the date you Paid it Online. Dates matter and need to match your reality.
"a few days before the downloaded transaction date."
Meaningless. What you have is proof that it actually and finally got to the bank and was processed. This date is Not Yours and you don't use it for your own financial records. It's an activity that you were hoping to see = verified that the EFTPS got processed. Now you will confirm that the Amount you entered into EFTPS is the same amount QB told you that you owed, because you Already have the entry in QB.
If you pay me by check in Nov 2018 and I didn't cash it until Today, you Spent that in 2018 and that is your 20148 business tax reporting expense. Banking Date is not Your perspective.
"how can I match this newly created liability check with the transaction downloaded in 2016?"
Is your Banking in QB fully reconciled? If not, then you have the EFTPS check, already. The "download" isn't even in the Banking Center, is it? If your banking is reconciled, you will Delete the wrong entry, because you just made it using the Right Function. It is no big deal to re-reconcile. You do all your housekeeping. Let's pretend that all of this is up to Dec 2017 and nothing beyond is reconciled, because you are Catching up all bookkeeping and taxes were never reported and are later. Once you fixed everything, you use the Dec 2017 bank statement's ending balance and date, you ignore the Beginning balance, since you know your housekeeping changed, deleted, and otherwise affected previously reconciled transactions. You Rereconcile to that ending balance for that date, and you see all the housekeeping entries are showing that you know previously were cleared, so you clear them now. You still get to a different of 0 and now you are On Track with the Replacement Transactions.
You can decide to Never Use banking downloads. They are a Helper function, to prereconciliation, as a way verify entries you made are on track = that is the Match function. And if your QB system, for instance, has entries differently, then there might not be a Match, but as long as the Amounts are right, you can mark the downloads selectively as Ignore = delete the download, I am not going to use it. Example: You decided to use Write Check to enter the Refunds to an employee but you wanted to list actual Vendor or Payee, and in reality, you wrote that employee one check. That means Banking Download won't be an exact match to Multiple Transactions but the Total is right. There are ways to use QB to prevent that, or you know the Values are right and you like your method = Ignore the one Check entry from Banking download. You still reconcile just fine. Or, I have a client that uses Receive Payment and wants each to post individually to the Checking account, even though she batches one large deposit. She won't use the method of Undeposited Funds to batch it, and she won't use the function the Add More Details, to match the batched deposit to individual open invoices. I won't fight her on this, because the Bank reconciles fine. Her method is crude, but it is Fine.
"In the future, I will create the liability check first before downloading corresponding transactions. I want to fix the existing problems first."
You don't use Download Banking historically. You manage your Banking. You make sure things are right, and Reconcile to each statement, working your way forward. You don't need Banking download at all. It's a Helper tool, not a requirement. Don't use a tool that makes More Work for you, for the historic housekeeping you need to do, now.
