Weight training in my view is crucial for health, especially as we age. Many say that if you want to improve your cycling, you just need to ride more, but that's just not true, despite it sounding totally logical.
Cycling is not a full-body activity, so you need to do something that exercises those muscles (and all other parts of your musculoskeletal system, i.e. connective tissues...) and weight training can fill in those gaps. Also, it's a total myth that if you cycle, you don't need to do lower-body exercises using weights -- cycling does not exercise all parts of the legs equally and can lead to muscle imbalances over time.