"Sludge" at the bottom of matcha/green tea latte
If you are making your drink with real matcha, that matcha is very finely ground green tea leaves. There are parts of the tea leaves that will always be insoluble in water, and therefore eventually fall to the bottom and create this sludge to which you are referring. The only solutions to that issue would be either 1) filtering the solution to remove the insoluble solids or 2) allowing the insoluble solids to settle to the bottom and the decanting the beverage above that sludge into another container. Straining is not enough when you are dealing with very fine particles, as the openings in strainers would just allow those particles to pass right through.

If, however, you are not using real matcha and it is completely water soluble (maybe a freeze-dried preparation from brewed tea) there will be no sludge. I would suspect that Starbucks is not brewing green tea with real matcha for every serving.