The reason I don't chase mementos is because I can only have one at a time, and I don't necessarily want to go through all my crafted weapons and decide which one looks best with the memento at any given time. Even if it's just 3-5, being able to stack a few of the same memento would be very nice, and would make me much more likely to grind them out.
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Maybe the ones that are connected to the rarer "seasonal" type events should let you stack a few, but things like trials and gambit are already in a pretty good spot in terms of frequency.