This card is bananas for Carson. You play it, then give two actions to other players and go check if the washing machine has finished. Let the fierce ones handle the probability calculations and drawing tokens while you enjoy the pleasures of servitude.
I think this might be an amazing card for Lola Hayes. The Hyperphysical Shotcaster is a neutral card, which means that Lola would be able to activate it regardless of the role she's currently in. It's also quite safe against her weakness (Crisis of Identity), so there is little chance the weapon would be discarded.
It is a Myriad card that doesn't interact with its other copies unlike most Myriad designs that came in TDE IE box. By including maximum 3 you can place a vow on each player in full 4 players party. Playing on lower player count, more copies let you play multiple stacked on one investigator in order to move more than 1 dmg/horror in a round. You can include just 1 copy to optimize deck space and it still works fine. I find this freedom to choose a quantity a neat design for this one.
Here are the chances of drawing at least 5 events in your 13 initial cards for the ability, depending on the number of events in your deck. This includes Painted World in the total of 36 starting cards (weaknesses are reshuffled). Also included are chances if 2 cards are added during the campaign for comparison. This is calculated using a hypergeometric distribution (appropriately assuming no replacement):
Number of Events: Chance of drawing at least 5 in 13 -> New chance of at least 5 in 13 if 2 non-event cards are added
24: 99.9% -> 99.6%
23: 99.7% -> 99.1%
22: 99.3% -> 98.2%
21: 98.5% -> 96.8%
20: 97.2% -> 94.6%
19: 95.0% -> 91.5%
18: 91.8% -> 87.2%
17: 87.3% -> 81.7%
16: 81.3% -> 74.8%
15: 73.9% -> 66.8%
14: 65.0% -> 57.7%
Hopefully this will help with some of your optimization.
Clarification: 'Engaging Enemies vs Being Engaged by Enemies: When an investigator engages an enemy, that enemy has also engaged that investigator, and vice-versa. There is no difference between engaging an enemy and being engaged by an enemy. Effects that trigger “after an enemy engages you” will trigger at the same time as effects that trigger “after you engage an enemy.”' - FAQ v.1.3, May 2018: