The way you phrased your question made us chuckle here at Mr. Smarty Plants. Please don't think we are being impertinent if we ask you ... could too much heat and sun kill an Englishman in your part of Texas?
So the answer to your question is yes, yes, yes!
English Ivy has become a very agressive invasive plant in many parts of the US. You can learn more about it at the website of the Center for Invasive Sepcies and Ecosystem Health. The vine can kill even large trees by climbing up the trunk and shading the branches enough to compromise the tree's ability to photosynthesize. The reason it had not taken over your trees must be that the dry hot conditions kept it from being too vigorous.
So of course we think you should remove the remaining ivy and replace it with a more suitable, native ground cover. You are not limited to traditional "ground cover" but can consider a mass planting of perennials, grasses or smaller shrubs that spread readily.
Texas A&M has published a list of recommended vines and groundcovers for Texas that you may find helpful. However, not all of the species listed are native to Texas. You can use our Native Plant Database to generate a list of plants that would be suitable. If you click on the North Central Texas map on the Recommended Species page and narrow the seach to include your conditions (sun to part shade and dry) it will generate a list (with links to more detailed information pages with images) of plants native to your area that are known to perform well in garden situations.
Here are some evergreen recommendations that were made for your area in a previously asked question:
Calyptocarpus vialis (straggler daisy) grows 6 to 12 inches and is almost evergreen. It does go dormant in cold winters in Dallas. Phyla nodiflora (turkey tangle fogfruit) is evergreen in Dallas in mild winters and grows 3 to 4 inches tall. Geum canadense (white avens) is evergreen and grows 4 to 18 inches high. Polystichum acrostichoides (Christmas fern) grows 1.5 to 2 feet and is evergreen. Salvia lyrata (lyreleaf sage) grows 1 to 2 feet tall and is evergreen. Packera obovata (roundleaf ragwort) grows 1 to 2 feet tall and is evergreen. Carex planostachys (cedar sedge) normally grows to 6 inches and is evergreen. Carex blanda (eastern woodland sedge) grows to 1 foot and here is more information. Carex cherokeensis (Cherokee sedge) grows to 1 foot and here is more information. Here are photos from our Image Gallery:
