A High Deductible Health Plan is a requirement to establish a HSA.   Funds put into a HSA are not taxable when used for qualified medical expenses, but there is no credit for not using or not having a HSA.

I could not find a question regarding the date of the HDHP.

There are no tax bonuses just for having an HDHP.

If you paid the premiums out of pocket, you might qualify for the premium tax credit, but this would apply to any health insurance.

An HDHP only results in a tax benefit if you also contribute to an HSA.  If you did not contribute, then your mistake did not change your tax owed or your refund.