Does your daughter do any classes/hobbies? In my social circles it would be very unusual by four not to do anything at all, and if your MIL is on another planet then my family and friends and colleagues must all be on the same planet with her. By four, I was doing dance (ballet, modern and tap and had already been in a few shows) and swimming which my parents considered essential as my Uncle nearly drowned as a small child on a day trip. I started both around age 2.
Among my friends, they all seem to start their children in activities from that age as well. Swimming is a must for all of them and they definitely all start long before school, both for safety reasons and also so they can enjoy the pool on holidays and as an activity locally. A few of them actually started swimming sessions as really tiny babies (I think it's called WaterBabies)!
Other activities their pre-schoolers are enrolled in are Rugby Tots, Diddikicks (football), Jo Jingles (music), TumbleTots, Yoga, Drama, Ice Skating, Acro, lots of different Dance based activities. I literally don't know anyone who has a preschooler who doesn't do at least one activity, and to be fair it's usually two.
If she's a primary school teacher she will have more experience than you in encountering lots of children and knowing what helps their development. If money is an issue and she's offering to pay for swimming, allow her to. I'd agree that 4 is a little young for piano but, providing your dd is interested, it's not too young for a group sensory music class. Languages...I think can wait. Really swimming and whatever else dd shows an interest or aptitude for, whether it's
sport, arts, music etc.
It's a great time to learn, their little brains are like sponges soaking everything up, good for social skills and confidence, understanding appropriate behaviour etc. I'm still friends with people I danced with and did shows with as a toddler. There's really no
harm in a child having a couple of organised activities per week. I definitely think there's potential harm in a four year old having no water skills.