I'm going to sound like a real w&@ker and killjoy here but if a secondhand lifejacket fails can you get a secondhand child to replace? Sorry a subject close to my heart. If you do go secondhand please try to get it serviced, I think the RNLI still offers advice/guidance. Apologies for my rant and mods I'll slap my own wrist.
Don't apologize for giving good advice.
A couple of years ago my sailing buddy and I went for a bimble on the Humber. We're both experienced sailors and when we turned up near the lifeboat station under the Humber Bridge looking like a pair of numpties the guys from the station came over to have a chat. They asked us about life jackets, flares, radios and other gear and when we showed them that we had all that sneaked away in the boat, they were quite surprised because it was a high-performance thing a bit like a two-man skateboard, with sails, but nowhere obvious to put anything.
Anyway, just before launch we started checking over the gear and Andrew (I'll call him that because that's his name

) decided to test inflate his life jacket using the mouth tube. The fabric of the life jacket didn't leak, but imagine our surprise when the whole tube came off in his hand when he started to let it down. Andrew had bought that life jacket new, it was less than five years old, and to us it looked fine until it broke. It was a plastic part that failed, not the fabric of the jacket.
Of course he'd have died if he'd had to rely on it. Now we weren't expecting to do that, even if we went in (and we _were_ expecting to go in, and we weren't disappointed

) but you just never know. Anyway after the smug grins we'd given the guys at the lifeboat station, imagine our embarrassment when we had to go and cadge a life jacket from them for the day. It was either that or go back home. They were really very good about it and hardly made us feel like idiots at all. We carefully washed all the salt off before we gave it back.
If it's a small boat, I'd recommend not getting an automatic inflating jacket. After a particularly wet beat into Walton one night on a much bigger boat, Andrew's automatic went off in his bunk while we were eating. The little bit of soluble stuff that triggers the inflation had somehow got wet even though it was folded inside the waterproof pouch in the jacket.
Probably better to get one that doesn't rely on inflation at all for a youngster, you don't know if they'll have the presence of mind to pull the cord if they go over the side, and certainly for one as young as three. At least if it's second hand and it's not an inflating one you have a better chance of inspecting it to see if it's likely to work, and perhaps of it working if it's needed. But old parts are still more likely to fail on anything.