I think you are getting a bit of an unfair kicking here, because people are ignoring the fact that you have a relationship with your son, independent of his wife. It's lovely that you say you love her - and it's clear the two of you are pretty close - and I think you have been absolutely gracious in how you dealt with and took on board her comments.
But there should still be occasional time for you and your son to spend time together.
When my DD was little, I was the main breadwinner with a time consuming out of the house job (SLT in a big school). My then husband worked from home so picked up a lot more of the day to day parenting.
My mom and I still spent time together, and my husband raised no objection - he spent time with his mom too on other occasions.
I think the perfect solution might be if you had taken your son AND the baby out? Then DIL gets a break and a chance to chill, and you get morher-son time.
But I do think the parent-child relationship still needs and deserves nurturing right into adulthood. My mom might be a grandma now but she is still my mom. And even if my DD one day has a child, which I would love (I really want to be a very hands on granny if I am allowed to be!) my DD will still be my one and only child and the most precious thing to me in the world......
Does that all make sense?
So please don't feel you can never have days out with your own son any more. But maybe, where possible, bring the baby, so that DIL doesn't feel put upon - I do understand what she means in that sense (I think she was cheeky to suggest you should fund dates for them tho - ask you to babysit, yes, but i dont think she has the right to demand you pay for her to go to sporting events!)