Most states require PFDs be worn for children under a certain age. This is for a good reason. This is often followed by reports that the children survived, but the parents did not.
The excuse that “I will paddling near shore” is an excuse that is often reported in a number of paddling fatalities.
On the water with a toddler? You don’t wear a PFD I don’t have much faith in your safety choices.
I have been hit by some large wakes from boats near the shore. Tipping in shallow water with a rocky bottom sucks.
I take my kids out on the water (when they are actually interested) and have done so since the youngest was 3. It can be fun and it can be terrible depending on the day 
Everyone’s experiences will differ but I have found we have the best days and the most fun when we use a SOT and an ancient barge sized open cockpit tandem. On the SOT my kids like to drag their feet in the water or just jump in and hold on so they can be dragged along… which is great for speed and efficiency In the tandem they are interested in one thing… snacks!
Safety wise, everybody wears a PFD or swim floaty, we do like to throw them off the boats and have them swim to shore so they are used to bobbing around. We also stick to a local river and lake that has minimal boat traffic and we never stray further from some type of land that they could swim to if they had to.
I did try using a sit inside once and turned around pretty quickly once I realised how dangerous a wet exit could be. It made a nice picture but I wouldn’t do that again.
Ultimately my kids like kayaking but it is a means to an end, it is a way to get to a beach and a way to get snacks. I wouldn’t go into it thinking you were going to have a nice long or leisurely paddle full of quiet reflection or appreciation ha.
We stopped using the puddle jumpers last year after reading some not great info on them. My kids are now using youth jackets from NRS that fit them great, plus they love having grown up looking jackets.
I think canoes are a better choice. Kids can sack out, sprawl, lean over side with a hand dangling in the water, sit in a chair, etc.
I took my son down the Missouri when he was 5 or so… he would fall asleep in his chair under an umbrella while I made miles.
I had my son sea kayaking with me, albeit briefly and close to shore in Monterey Bay, from the age of 18 months. You’ll note, he is/was wearing a PFD.
When he got older, we used to rent sit-on-top kayaks from Monterey Bay Kayaks for excursions much further afield. Great fun!
Start your kids early, never look back, and ignore the worry warts who are overly cautious. I imagine the drive down and back home was far more dangerous for him than any paddling on a calm ocean, on a beautiful, sunny day.
I also had him snorkeling with me from age 2, in Carmel Bay.
I will add that I am a very experienced swimmer (competitive), freediver and scuba diver, lifeguard, Water Safety Instructor, and sea kayaker with lots of experience in and on the water.
Also, as pointed out, kids can learn to swim very early. I’ve taught 10 month old children to do that. My son was swimming at about this age - 18 months, or so.
Be careful, but go forth and have fun. You’ll make great memories for a lifetime, if you do that.
If you look closely, you can see my son in the cockpit with me.
You have infected him with kayak fever! I have a photo somewhere of our 1yo grandson with my first carbon fiber paddle. He is 21 now and planning to kayak fish at the beach next week.
The question to me is can your young passenger handle the water alone if something happens to the adult who brought them out there?
Have the child wear a PFD (probably required by law anyway). Always wear one yourself. It may be critical in helping you to help your child.
Have fun!
