Respectfully, I see that you are trying to hold the following objective:
Strategize for Winning by Actively utilizing Gadgets
My critique is that the proposed 4 Rules meet this objective while simultaneously introducing other parameters that limit creativity (which, ironically, seems to be the purpose of the format). Looking at each one:
1) Your deck must contain at least 12 cards with the gadget keyword.
This seems fair, but by itself wouldn't preclude archetypes that "splashed" gadgets (or simply "filled out" hearts) for the sole purpose of meeting this requirement. Hence I can see the need to include more than one rule...
2) Dudes without a gadget or the gadget keyword cannot be chosen to participate in a posse (they can still be called out :D)
3) Dudes without a gadget or the gadget keyword have -1 influence (minimum 0).
These rules heavily skew incentive to include cards like Prof. Aloysius, Yagn's Exoskeleton, the various Gadget dudes (Quarterman, Postatron, Wretched), and cards that tutor Gadgets like Maggie Harris and Technological Exhibition. And while these are all very "gadget-ey" cards, by consequence all other "gadget-ey" cards are categorically disincentivized by virtue of having limited interactivity with these two rules, which incidentally limits creativity.
A second point is this creates a new dynamic within the rules that changes two of the fundamental aspects of the game: Game Clock (or "relying on influence to stay in the game") and Chess (or "using callouts / shootouts to move opposing dudes").
4) You score 1 point for winning a match IF you have more gadgets on the table than your opponent by the end of the game.
This rule (especially backed by #2 and #3) favors Straight-flush in Hearts structure above all others.
In conclusion, this is anything but a casual affair, because the liklihood that a gadget deck can hold it's own in a conventional game of DTR has a skewed bearing on whether or not it could hold it's own in this format - almost like two different games.
One question: Do you have to keep using the same deck between games?