How do t-lymphocytes become mature if thymus has lost its function?
A few points, I'll expand on this later and add references if I have time:


The thymus starts degenerating at puberty but continues to function in thymocyte maturation for decades, albeit at a lower level
Many mature T cells can live for years
Through homeostatic proliferation, peripheral T cells maintain their population size
Immune surveillance generally decreases with age


Note that extended lifespans and homeostatic proliferation (points 2 and 3) don't contribute to TCR diversity but only maintain the existing repertoire.