How to raise the overall effectiveness of an aviation maintenance company regarding productivity and quality?

Do your maintenance by the book and on schedule. I have seen/heard pilots and Production Control guys argue over the wording of regulations regarding pertaining to flying aircraft past their required phase maintenance windows. Just do the maintenance when you're supposed to, and you won't have to worry about "interpreting" overflying maintenance regs. Consult with your TIs, as they will give you the book answers, and how they want to see the work done, as they are the ones who will sign-off your work. They also have the power to ground one aircraft or the entire fleet. I've seen that done when companies wanted to overfly maintenance windows. Grounding the fleet quickly brings a huge spotlight to the maintenance issues, as well as the attention of the BN CDR.

My first thought would be to make sure people are given the time to do maintenance, properly supervised, and not "drawn away" to do other duties that take them away from doing their maintenance jobs. "Cutting corners" to make up for lost time doing other activities generally costs more in shoddy work that has to be re-done and inspected more.

It is always best to show a link to the work that needs to be done and the reason it needs to be done professionally. For example, I used to explain to my sailors when I was an Engineer that if our plant fails in combat, all of our buddies on the ship are in danger (as well as ourselves). In your case, those airframes need to be in the best possible condition to let the pilots take them into combat and protect other service members.