Frontsdoor, toddlers are being offered the flu vaccine to protect themselves too, it's not all about herd immunity, or protecting other family members.
Seasonal/endemic flu is a serious risk to small children, far far more risky than the risk of vaccination.
One thing to remember about side effects, more are not "side effects", they are "effects" that show the immune system is reacting well to the vaccine... Sore arm/ localised swelling and pain (if its an injection), feeling a little under the weather is normal after vaccination.
More serious side effects are rare, and info is collected on people who develop them - it doesn't mean the vaccination is to blame, most will be coincidence, some of course will not, but the risk of that happening is tiny compared with having serious side effects from catching flu (I mean real flu, not a cold).
My 3 year old had his yesterday, but all (4dc and 2adults) of us get the seasonal jab every year (no particular risk factors except for DH due to his job). As a GP he has seen what flu can do, to all age groups, it's not just "old people" who are at risk, but many people seem to think that.
People are not very good at assessing relative risk! Read the scare story websites by all means (they do raise interesting questions sometimes), but for goodness sake then go are read some actual science before deciding whether you want to protect your chile or not.