Today if I were doing that I would cut it down, would chop (use a mattock and have on steel-toed boots) at the main roots and take out as much from around the main bole or stem. You may have to cut away new growth for a while, but it will eventually die, and then you can drill holes in the wood and sprinkle dry molasses over it (or pour over regular molasses) so that the sugars help the wood break down faster. At my current house it took a couple of year for the roots to break down from that front hedge, and I think the application of sugar would speed it considerably.
I'm sure there are many other approaches.
I hope you're going to leave the crape myrtle in the front?
Thanks for the tips, and yeah, we'll probably leave the crape myrtle, just really want to get rid of the hollies mostly.