1c8e3b8a6fd64597aefd213dcda88bff
Why does BiB does not show Ashes of the Embersoul (493) as better than Pip’s Emerald Friendship Badge (476) even though when clicking on the item in the list clearly shows the Ashes as adding DPS?
The ranked lists are ranking each item “in a vacuum” - as if no other gear changes and you simply swap out that one item. Once you score it with an entire set of gear, the results can sometimes be slightly different.
Especially when you are talking about relatively small differences, this can happen from time to time. Trinket rankings, in particular, can be quite difficult and sometimes even a bit subjective. If you really like the on-use of the ashes trinket, you could lock it in or increase the value of it in the customize tab.
On-use trinkets can sometimes have more value than the numbers suggest, depending on what content you are doing and how it times up.
Well, I just used my Bullion to buy that, thinking it would be a net increase 
Are the upgrades, bullion, etc lists the same? i.e. scoring ‘in a vacuum’?
Yes, the upgrade finder scores each item in a vacuum - it would be cost-prohibitive and quite slow to do a best in bags for each and every item in the list. You can use the “add to my bag” feature to do a full BiB optimization on a small list of specific items.