I'm the player of that character in knikpiw's long-running game. My android started at grade 4 with LOG 10 (thanks to the +2 from the Service Droid origin) and got a +1 from every advance, I think. I ended the campaign at grade 10 with LOG 16. I think I would have hit grade 11 if we'd gone one more session.
Around grade 6 things felt pretty good, since I had maxed (but not exceeded) my pools in the areas I was specialized in, and could get up to the cap in some less specialized fields if I had high-quality tools (which I did for a few things). But I do agree that there was never a mechanical reason for me to take my skills up to 6 ranks, though I did so for in-game reasons (we'd been offered us jobs that required somebody in the party to have 6 ranks in certain skills to qualify for). Without those non-mechanical pressures I'd may not have gotten any skills that high, certainly not for LOG based stuff. Though once I had them, I certainly wanted to push for more grades, so my MDP would go up again and I could use the dice I'd earned. It just took a really long time to get from grade 8 and a 7d6 cap to grade 11 where I'd have gotten an 8d6 cap.
I also feels like it is a bit of an awkward design that you want your highest skills to be ones that you have bad attributes for. A thief probably doesn't need six ranks in stealth or thievery because he's going to be super agile already, and might be better served by having lots of bluffing. A scientist doesn't actually need a doctorate, because having lots of career grades that will increase her LOG means a lower degree is going to be good enough to get to the die pool cap (in a well equipped lab), while expert marksmanship training with pistols could come in quite handy surviving in the field. I'm pretty convinced that there is never any good reason to take a skill higher than six ranks, and even going from three to six is of rather marginal utility. If it's for an attribute you're bad at, it may be easier to buff the attribute instead of the skill, and to get high quality gear to fill out the last few dice before the cap.
The pregenerated character Dr Tiktok is a good example of the problems a specialized character can face with the MDP cap. Thanks to his doctorate in medicine, at grade 5, he's got 6d6 in his LOG (medicine) pool, exceeding the 5d6 MDP (and he has even wasted one skill increase getting rank 7 in medicine for no good reason). If his next career advance gives him +1 LOG, he'll be exceeding the cap again because even though the MDP goes up to 6d6, he'll get an seventh die for LOG (medicine) checks when his LOG goes up to 10. (Speaking of attributes, that character has perhaps the most aggressively suboptimal starting attribute distribution I can imagine, with five 5's and a 9. Get one more +1 in any of those attributes and you'll have an extra die in your pool!)
I think a possible solution might be to let each character have one skill that is not limited by the MDP cap. Perhaps it should be related to their primary career that they've included in their descriptor? You could swap out the skill each time you took a new grade. This would allow a specialist to really shine when their focused skill is relevant, without letting them dominate a broad range of tests just due to one high attribute. It might be necessary to prevent [combat] skills from being selected for this (just like you can't use your hook to make die rolls explode when you're in combat).