True. After mid July, the stems contain more and more oxalic acid which can cause stomach upsets and exacerbate arthritis and gout in susceptible people. The advice is always to stop picking by end of July which also allows the plants to build up reserves for next year's crop. However, I just picked my last crop three days ago. :-D
I'm about to cut the last of mine also. I always leave a few stalks on to build up the reserves for next year. A sack of manure also wont go amiss as well.