I think that Luke Robinson is an interesting investigator. I'm not sure about his power level, but starting in play with such a powerful asset greatly increases his potential during the game. Gate Box essentially gives him 3 uses of Elusive which is super powerful given that Elusive costs 2 XP with the Taboo List, so he basically gets 3 uses of a 2 XP card for free. To be fair, the effect he gets is a little worse than Elusive, as it actually costs an action to move to another location, while Elusive is actionless, but if he waits until the end of his turn, he automatically moves away for free. But that's already great power level anyway, so the action cost can't even count as a disadvantage. And what's more than that, he can add even more uses to his Gate Box with Enraptured and Recharge making that asset even more powerful!
His deckbuilding options are a bit weird, but interesting. Seeker class for a 3-Intellect investigator isn't great but offers him interesting options. I think he might fall in the same category as Roland Banks when it comes to investigating, arming himself with Flashlights and Working a Hunchs and clearing low-shroud locations with regular investigations and using auto-clues for the harder stuff. He can always use Rite of Seeking and Sixth Sense of course, but if he wants to use more physical means for investigating, I think his options are kind of limited.
His ability is also interesting. I think the engagement part is mostly useful for multiplayer, but he can generally equip himself with the right events to make it count. Banish, Storm of Spirits, Drawn to the Flame and Working a Hunch make some examples of some options he can include to make use of his ability, and I think he should also lean on events heavier in order to make it more worthwhile. What's interesting here is that he can combine his ability with Gate Box. Dream-Gate is connected to every other location in play, so he can use the Gate Box to move to Dream-Gate and deal with some enemies or get some clues from any location he wants and then move to any other location he likes! I think that's super neat!
Luke Robinson's weakness is relatively mild in my opinion. With his 9 sanity, I think that the 2 horror it deals to him is a minor punishment. It can even benefit him in some situations, by letting him move to any other location he wants. I believe the worst part of it is that it can lock you down and detach you from the rest of your team for a turn and that can be very punishing in certain situations. That's, of course, assuming that he can't take a successful 6-difficulty investigate test, which I believe will be the case most of the time, but if he somehow can, he can mitigate this problem. If he runs Mr. "Rook", though, he can use this weakness to his advantage, as Mr. "Rook" will give him more control of when he draws it and possibly turn this downside into a benefit, by letting him draw it at an unimportant time and allow him to move to any other location he wants for free.
As a final note, I think De Vermis Mysteriis becomes a lot better with Luke Robinson. All those mystic spells that fight or evade can now be used against enemies at a connecting location and let you avoid the attack of opportunity that De Vermis Mysteriis would cause. Correct me if I'm wrong, but I believe that the cost is paid BEFORE the ability is initiated, which makes De Vermis Mysteriis usuable while you are at a connecting location and when you initiate the ability you can choose to use Luke Robinson's ability and play the event as if you were at a connecting location, therefore making it usuable with events that fight or evade enemies while you are not engaged with anything. And, as a bonus, his combined access to Mystic and Seeker cards gives him good options to fuel the book, making it possibly a viable choice for him.