What I see is Not Correct. Let's review:

You charge a Service, so that is linked to Income. This will control the flow of the data to accounts.

You use it on Sales Receipt = the date of the sale is also the date of the payment.

Or, you use it on Invoice = This is Income, but not yet a Paid sale. This creates the AR. Your Item is not the AR. The function of using Invoice creates an Unpaid sale, which is AR.

Then, you get paid; this is Not a Deposit. You should never see DEP. This is a Payment, when you used Invoices. Or, you already got the funds from the Sales Receipt. The only time you would see DEP is if you Batched all the payments and your sales receipt by sending the payment data to Undeposited Funds, to be held until the banking shows up.

So, it is like this:

Sales invoice (income here) ==> AR ==> Payment arrives ==> Held as UF ==> Banking happens

That's why DEP never shows Engineering Fee Income. That is Settlement of the sales, and not your Income event.

Let's break this down a bit:

"At which point QB puts that money in A/R."  I wish it didn't, I wish it could just corroborate with the bank/Paypal deposits."

It is Neither AR nor Banking; you need to edit your Charge items and link them to Income and let QB move existing data for you.

"Worse still, these transactions are hidden in my A/R, as my P & L reports don't see them as income."

You will always see Invoice as the Transaction Type that creates Income. But the income doesn't show on Cash Basis for unpaid sales; it shows on Accrual Basis for the date of the invoice, even if Never paid. For cash basis, it shows that invoice for the date of the Payment, on Cash Basis, but you still see the Invoice. The invoice (the Sales) is the Income transaction.

"I wish it didn't, I wish it could just corroborate with the bank/Paypal deposits."

If you use the banking download, never let this Add to the file; these are not DEP, if you never yet processed payments. They are Matched to open invoice. Or, Matched to the amounts still sitting in UF.

And you should be managing PayPal as its own Bank account in your system, if they hold your funds and you incur fees there. That way, a sales receipt or Invoice Payment would post to the PP bank, not to US, because they are posted Individually in reality. Bookkeeping matches reality. Then, settlement to Real Checking from PP is a Transfer; this is not income again. It is you, moving your own funds around. Your PP fee is a Check from the PP bank. You use the PP statement to reconcile your PP bank in this file.

"P.S. I use the cash method (I tried accrual but it seemed more problematic and unnecessary, I'm a sole proprietor)."

You run the Reports on Both Basis, to keep on eye on things. You are using invoice to Anticipate payment will arrive, because you have in fact charged the people. That is fine.

Cash or Accrual is a Tax Reporting perspective. In the accounting, we use the tools that meet our needs.

I have attached some images that are from PC desktop, but they should be helpful to you.