Hello Jay,
By all means...if you do not like Ivy (Hedera sp) an introduced species, you can cut at base (carefully) there by weakening and/or killing the plant. Goats have been used effectively as well.
I would note that most "climbing vines" do not kill the "armature true" usually or even weaken it...unless the tree is very old and already stressed. There are tropical Ficus and a few others that are parasitic and kill host armature trees. I would also not that even old architecture is not directly damaged by vines in most cases.
I feel one must way not only personal aesthetics, but habitat loss.
If the "looks" of ivy aren't of value to a client, I just monitor the grow patterns and perhaps prune and shape to the desired perspective.
Regards,
j
FYI: if interested
here at Ivy.org.