I was working on a system for controlling ships that dealt with similar issues:
On a sailing ship, the First Mate decides how far the ship goes in the round by commanding the crew to "trim the sails" and "batten the royals" and "jib the mizzenmast" and other silly nautical terms, and can make an ability check for the ship to exceed its normal maximum speed. But the ship doesn't actually move until the pilot's turn; the pilot can only take a single 45-degree turn during the movement, unless they make an ability check to get more turns. If nobody pilots the ship it just plows ahead at the previous turn's speed on initiative 0. Sailing into the wind was more or less difficult depending on whether your ship had square or lateral sails.
Rowed ships (galleys, longships, etc.) were the same way, except it was the Boatswain who would determine how far the ship goes, by making the rowers go faster or slower. Ships with both rowers and sails could use either.
Small ships under manual power, like a canoe or poled barge, could have multiple people trying to drive and steer them. For example, the two people paddling a canoe could each cause the canoe to turn up to 90 degrees and move up to 20 feet -- except the canoe's total movement for the round couldn't exceed 30 feet.
If I were building the Mobile Combat Platform of Kwalish, I'd probably phrase it in those terms as well, instead of levers. For example:
- A character within 5 feet of the engine can use their action to alter the platform's speed, increasing or decreasing it by 20 feet, up to a maximum of 40 feet. The engineer can make a DC 15 Intelligence (tinker's tools) check increase the speed by 40 feet, to a maximum of 60 feet, until initiative count 0 on the next turn.
- A character within 5 feet of the steering controls can use their action to cause the platform to move up to its remaining speed, turning during the movement. The platform can only turn up to 90 degrees each round, unless the pilot can make a DC 15 Dexterity (land vehicles) check to cause the platform to turn up to 180 degrees.
- On initiative count 0, losing ties, if the platform has movement speed remaining, it uses that speed to move forward. The remaining speed, remaining turning, and claw attacks reset.
- A creature within 5 feet of a claw control can cause one of the platform's claws to attack a creature within 20 feet of the platform: +8 to hit, 2d6+6 bludgeoning damage, or the target is grappled (escape DC 16). Each claw can only attack once per round.
The idea here is to cap the Mobile Combat Platform's capabilities, rather than the characters'. For example, nothing prevents two characters from both piloting the platform in the same round; but during that round, the platform can't turn more than 90 degrees, and moves exactly its current speed.