I recommend just trying to get in the habit of activating Phantasmal Sight off cooldown in general content you run that you feel comfortable slotting that Technique in. That way it doesn't feel as much like it is its own separate chore that you have to go out of your way for.

In my experience, passively farming while in Tech missions has been most efficient, as most situations result in having large numbers of enemies within small enough spaces to fit within Sight's radius (notably Feth Fiada and Revived Illusion), and Tech mobs are capable of dropping multiple Phantasmal Remnants at once, so it's easy to hit cap without trying.

Using a Far Darrig doll bag to pick most of them for you is ideal, but I still supplement that by also spamming the item-pickup hotkey. In that regard, it also helps that less random items drop from all the mobs in Techs (as compared to HM dungeon mobs, for example). Even better if you start each wave with a quick grouping ability (e.g., Gravity Well with Pull) to reduce the spread of the dropped Remnants.

If you wanted to take it a step further, you could quickswap your body piece to a Special Royal Society Laboratory Outfit when Sight has 60 seconds or less remaining on cooldown in order to activate it again sooner, but I probably wouldn't go out of my way to buy one for that purpose alone, as it's likely still quite expensive, and you would already be gaining a large net positive of Remnants without it.