Zero Vs Fixed Coupon Bond volatility


I am new to the forum and I probably come with a very easy/obvious question but I can't seem to get my head around it.

Everywhere I look for this on the internet I read that Zero Coupon Bonds are more volatile than Fixed Coupon Bonds. The explanation for this seems to be that the Duration (both Macaulay and Modified) are larger the lower the size of the coupon is, and consequently a, let's say 10Y ZC Bond has a larger duration than a 10Y 5% Coupon Bond. So, the larger the duration, the more sensitive it is and the more volatile it gets.

Ok! All fine with that... until I think about DV01. My background is derivatives so I've always "struggled" with the use of Duration as a measure of sensitivity rather than the usual Greeks. In DV01 terms, surely a $100 Notional 10Y ZC Bond has to have a smaller Delta than $100 Notional 10Y 5% Coupon Bond. Without getting into technical details, I intuitively think:

Bond_DV01(5%, 10Y) = CF_DV01($5, 6m) + CF_DV01($5, 1Y) + CF_DV01($5, 1Y6m) + .... + CF_DV01($5, 10Y) + CF_DV01($100, 10Y)

The last term (CF_DV01($100, 10Y)) is the same as Bond_Dv01(0%, 10Y) which is the DV01 of a ZC Bond. So, in other words, the DV01 of a 5% Coupon bond is the DV01 of a ZC Bond with the same maturity plus some additional addends with the same signage, so Bond_DV01(5%, 10Y) > Bond_DV01(0%, 10Y)

So, if we agree the DV01 of a ZC Bond is lower than that one of a 5% Bond, how do we justify the first statement which is repeated all over the internet (i.e. ZC Bonds are always more volatile than Coupon Bonds - with the same maturity of course)?

Looking at DV01 for me the obvious answer seems to be the opposite.

Anyone has any thoughts?