I agree with you. I have a group with whom I do the harvest temple weekly but in that group I usually run quickness scrappper to provide quickness, so not much room to play another class there. Yes I know other classes can also provide quickness but having to learn the whole rotation for just two times I need to play it in a specific strike mission is not very motivating.
Last Tuesday I tried to pug it on an elementalist my second most confident class and it went okayish sadly did not get the kill. I am sure I can get it done on some classes but also not looking forward to having to do it on others. I think a good mechanic would be that in order to get all the trinkets you have to complete the harvest temple 18 times with whatever class you prefer. Still twice for each class but no longer tied to the specific class. You could consider making adding a third kill if you are not using the designated class to still give players an incentive to play it on different classes.