Cross training vs mma training is a an ongoing never-ending debate.
It all depends on the instructors, imo.
Cross training is great for depth in every single art, but you will also spend a lot of time learning a zillion things that don't apply to (look at my killer collar choke) or might be dangerous in (my deep half guard is amazing) MMA. You will also receive contradicting information: wrestling coach tells me to hunch over but MT coach wants me to stay more upright, boxing coach says my lead foot tip must point inside but kickboxing coach says point forward (of course, these examples are generalizations). Then, you will have to go the extra mile to decide what techniques to apply, how to adapt them and finally jump into MMA sparring anyway. Some coaches may be more willing to help in this area, but it's not common.
On the other hand, MMA training can give you a more consistent approach offering an all-in-one package even if classes are distinct, because each art is (or rather should be) taught with the goal of putting it all together. That makes life a lot easier. However (big however), finding an MMA gym that has truly competent coaching might be a crap shot.
Of course it's an endless debate. Just going off my own personal experience.
I definitely agree it depends on the instructors. But I think it's important to note that most MMA instructors will not give you the depth of technical instruction that a striking coach or a grappling coach will give you. What MMA instructors will be extremely good at is teaching you how to adapt/apply techniques (they are familiar with) in striking/grappling in the MMA sport context.
There are disadvantages like you touched on - a striking or grappling coach depending on the art will teach you contradictory things and they will not teach you how to adapt techniques to MMA. But I think these are conditional disadvantages - a striking or grappling coach will teach you contradictory things if you are cross training in a multitude of striking/grappling styles - so this is easily remedied by learning one striking style & one grappling style only. Then adding further when you are competent in both. I wouldn't advice anyone either doing MMA or something else to engage in more than one grappling style or more than one striking style - especially if you're a newbie.
As for a striking/grappling coach not being able teach you how to adapt techniques to MMA - sure this is an issue. But you have to weight up the pros & cons. What a striking coach or a grappling coach will provide is indepth structured technical instruction - that will give you a good foundational grounding in the striking or grappling - like a striking coach will teach you proper footwork, distance management, angles, balance & weight transitioning during techniques, how to leverage properly with punches or kicks etc - all of which are transferable skills. A grappling coach will teach you basics in posture, positioning, grips, how to manipulate someone's weightetc etc things that most MMA instruction classes will not really delve into the same way.
This type of bottom up technical instruction tends to be lacking in MMA classes because time is divided between striking/grappling & instructors themselves have specific areas of expertise. The result is you will get rounded instruction but with no technical expertise in either area & you won't have the basic fundamentals down that someone who did striking or grappling separately will.
Habit is a very underrated thing in martial arts and once you learn something long enough it's very difficult to undo. Personally I think it's more important to learn basics properly rather than have gaps - because once you have those gaps it's very difficult to remove after it becomes habit.
I think that's probably the reason why most professional fighters in MMA learn arts separately for the most part & use MMA classes to put everything together.
It also depends on your goals - if you want to compete in MMA. I'd say you need to learn striking/grappling separately & once you have the basics down - then start doing MMA classes.