These old suckers at the base of our tree are thick and tough and so hard to remove with loppers or my small battery operated chain saw. They continue to spread each year. 

Unfortunately, there is no magical way to get rid of those suckers.  They are probably coming from below the graft and now that they've started, they will continue through the life of the tree.  You just have to persist.  Try using a hand-held pruning saw and see if that works any better than the loppers and chainsaw.

Here is an article from the New Hampshire Extension which addresses sucker growth on trees:

https://extension.unh.edu/blog/can-water-sprouts-and-suckers-be-prevented-trees