(Macaulay) duration is the
time weighted sum of cash flow
present values normalized by dividing by the bond price.
https://www.fincash.com/l/basic/macaulay-duration What you were using was modified duration (or effective duration), i.e. sensitivity to interest rates:
TIPS have higher durations than Treasuries of comparable maturities, so they are hit worse from rising rates.
Modified duration is the derivative of present value (PV) with respect to rates (again, normalized by dividing by bond price, i.e. PV). That turns out to be Macaulay duration divided by (
1+r) where r is the discount rate per coupon period.
This is easy to see. Start with the PV formula:

After dividing by the bond price, differentiating with respect to i (rate) gives:
{[(-
1 x PMT
1/(
1+i)&sup
1;) + (-2 x PMT
2 /(
1+i)²) + ...)] / BondPrice} / (
1 + i) =
- ( timeWeightedCashFlowPVs / BondPrice ) / (
1+i)
- MacaulayDuration / (
1 + i)
Related to, but not the same thing as Macaulay duration.
Still, that doesn't address your more significant assertion that TIPS' duration (whatever the form) is longer because cash isn't paid out until maturity. IOW, that TIPS are effectively zero coupon bonds.
With a traditional CD, interest compounds at a fixed rate. So calculating APY and YTM is easy. In fact, all that really matters (except for tax purposes) is the final value of that CD. You could call it a zero since you don't get the cash flows until maturity.
Still, there are interest payments; you can see it in the balance reported for your CD. The risk with fixed rate CDs (as with zero coupon bonds) is that interest rates may rise and you can't deposit those interest payments at the new higher rates.
If the bank did allow you to draw the interest payments and redeposit them at higher rates, that CD would be more valuable to you. It's not that you're literally getting your hands on the cash, it's that you're able to get current (higher) market rates on the interest as it is credited.
Same with TIPS. You don't get your hands on the inflation adjustments. But you see them in your balance (i.e. "principal amount"). And if inflation rates go up, that new balance benefits from the higher rates.
In this regard, TIPS work even better than redepositing the CD interest or reinvesting bond coupon payments. With the CD or the fixed rate coupon bond, only the interest payments receive higher rates going forward. With the TIPS, the original principal (as well as the inflation "adjustments") receive the benefit of higher rates.
With respect to inflation, TIPS are floating rate bonds, and as such have zero duration.
I started with the statement: "The relationship between inflation adjusted (real) durations and nominal durations is somewhat complex." This may help (or further confuse):
Nominal bonds are generally considered to have one duration (the sensitivity of the bond's price to a change in its nominal yield or interest rate), but inflation-indexed bonds, such as Treasury Inflation-Indexed Securities (formerly, Treasury Inflation-Protected Securities, TIPS), may be regarded as having two durations: Di, the sensitivity of the bond's price to a change in inflation, and Dr, the sensitivity of the bond's price to a change in real interest rates.
For a nominal bond, whether a change in yield was caused by a change in inflation expectations or a change in the real interest rate does not matter; the effect on the bond's price is essentially the same either way. But for a TIPS bond, an increase in inflation does not affect the bond's price because the change in the cash flows in the numerator (of the equation for discounted cash flow analysis) is indexed to inflation and the discount rate in the denominator has also been increased by the same change in the expected inflation rate. Thus, the TIPS bond has an "inflation duration" of zero. A change in real interest rates, however, affects the price of a TIPS bond much as it does the price of a nominal bond, so a long-term TIPS bond has a long real-interest-rate duration—say, 15 years.
https://www.tandfonline.com/doi/abs/10.2469/faj.v60.n5.2656That is why Vanguard moved to short term TIPS.