I had about 6 months of lessons in primary school grade 3 on a chord organ my parents bought for me (couldn't afford them electone things!!) ~ and that was it.
The rest was "follow the lights" on a Casiotone MT-800 decades later (in high school) as well as getting back to figuring...
Wow.... incredible story.
I wonder if something tactile like the seaboard might be something for you?
Unfortunately I am out of ideas - can you try the last one - raise the output volume of the keyboard (on the keyboard itself), and lower the "level" on the mixer your keyboard plugs into, Let me know if it makes a difference.
Also - does the keyboard have a headphone output? If so please check...
I just realised - if they can't/won't adjust the gate - try upping the output level on the keyboard (and obviously lower the level at the mixer).
If the gate is the issue, you need the "sustained" part of the note to be effectively (much) louder than the background noise.