First, you are correct that your employer does not owe you a corrected W-2.
It appears that the HSA custodian misidentified the return of the contribution, since a "contribution correction" should be initiated by the employer, not the employee.
When you enter your information into TurboTax, TurboTax will announce that you have excess contributions - hopefully the same amount that you actually withdrew. In this case, TurboTax will ask you if you will withdraw the excess by the due date of the return (April 15th). Since you already have, answer "yes".
That will cause TurboTax to automatically add the excess to Line 21 (Other Income) on Schedule 1 (Form 1040), so now you don't even need a corrected W-2, because the amount of the excess is correctly being taxed.
What's missing is the earnings from the excess contributions while in the HSA. If you had called the HSA custodian in early 2019 (i.e. now), then in early 2020, the custodian would have issued a 1099-SA with a distribution code of '2' and the earnings in box 2 - yes, the earnings would be reported on your 2019 return (because in this case, the earnings span parts of 2018 and 2019).
But we don't know if the HSA custodian will ever report the earnings to you. Given that you have no way to know what the earnings were on your own, I would say, just leave it be.
So long as the excess reported by TurboTax matches what you actually withdrew, tell TurboTax that you will withdraw (easy since you already have), and file.
And keep notes in your tax file of what you did and what the custodian did in the unlikely event that the IRS writes to ask you about this.