Hi there,
I can’t comment regarding responsibility – but I did have a query. Have you personally checked with Council regarding removal of the trees? If they are posing a hazard to users of the path, then they may be flexible. Perhaps you should contact them and request someone meet yourself and the arborist onsite for a discussion. Even if you have to pay for an application fee to get this done, you might be able to side step some of the potential legalities if you can get Council to agree.
Some Council’s will allow tree replacement also, i.e. removal of the offending tree provided a replacement is planted. Perhaps this can be mentioned, and the arborist can recommend something with a less intrusive root system.