Yes, we can say that the Theravada and Mahayana traditions have both preserved the early texts. But be careful about sliding from speaking about texts to speaking about traditions. Theravada and Mahayana traditions are lineages of teaching and practice, embedded within specific cultural and geographical locations. Those traditions have preserved a wide variety of texts, some of which stem from the early period. But not everything in Theravada is early Buddhism, nor is everything in Mahayana.
It’s misleading to speak of a “Mahayana canon” except in the limited sense of “texts preserved in Mahayana monasteries”. The canons in Chinese and Tibetan include a vast range of material, some Mahayanist, some not. I have a copy of the Bible in my small pile of books here, but that doesn’t mean that I am a Christian. All we can say is that, in the libraries of Mahayana monasteries in China, texts of early Buddhism were preserved alongside texts of later traditions, including Mahayana. This tells us nothing about the philosophy and practices of Mahayana, except that, well, they were good at preserving books!
If we want to look at the actual relations between early Buddhism and later Mahayana texts and philosophies, we must start by identifying and studying carefully the early texts, and how these were adapted in the Abhidharma traditions.
Then we have a good start at being able to see how the Mahayanists, starting at least partly as a critique of Abhidharma and the contemporary Buddhist culture, formulated their new texts, primarily as written documents, taking advantage of new technologies and social conditions. The Mahayana sutras, at least in the beginning, and probably for a long time, made no pretence to establish a new school, but to offer a new reading on old ideas.
Then the key becomes the interpretations developed by the central figures in the emergence of mature Mahayana, especially Nagarjuna, Vasubandhu, and Asanga. Each of these was deeply versed in the teachings of the Agamas and the Abhidharma, and each in their own way developed sophisticated approaches that unified the various aspects.
Finally, we can look at how these Indian traditions were adapted and transformed in the new homelands of Buddhism in China and Tibet.
Obviously this is a complex and difficult task. History is hard. But we don’t make it easier by throwing out the established facts.