If we are adding back D&A in CFO since it's a non-cash item, why don't we subtract the tax shield we are getting from D&A that is baked into net income, which is the first item of CFO?

Short answer is that D&A is a non cash expense and taxes are in cash. You don't subtract the tax shield that actually affects cash which as you said, is baked into NI. I can elaborate if you still are having trouble with the concept.

Can you please elaborate? Why do you get the benefit of the tax shield (via net income line of CFO) when you are already adding back the pre-taxed amount of D&A expenses?

So the cash flow statement is trying to show cash flows. The way most CFO's begin is through net income and adjusting for non-cash expenses. To illustrate, lets take a look at how we calculate net income, and why we add back D&A but not the tax shield.

EBITDA - DA = EBIT (We subtract DA because it is an expense that we incur but no cash his actually paid for this DA)
EBIT - I = EBT - T = NI (Interest and Taxes are paid using cash)

The cash flow from operations begins with NI, but NI does not truly reflect the cash we got or paid out so we need to adjust for the expenses / non-cash items that we have to determine the true cash inflow / outflow from our operations.

NI + D&A - increases in NWC + other non cash items = CFO

Tax is not non-cash, you eventually have to pay actual dollars out of pocket to the government.

Your understanding of why you add back D&A based on adding back "pre-tax" D&A may just be you not fully understanding the concept of CFO because there is not a pre/post tax D&A. You just have D&A.

Not sure if this will help

Depreciation is tax deductible and therefore provides a tax shield on the income statement. The full amount is added back as it is non-cash.

When arriving at net cash flow to invested capital the "tax shield" is removed from the interest expense when it is added back. I believe this is because the tax shield is picked up in the WACC using the after tax cost of debt. To not remove this would double count the tax shield.

On the contrary there is not such adjustment warranted with depreciation or amortization.