Freecycle it, or take along to a parent/child group and ask people to help themselves - it will go to somebody who wants and appreciates it. On Freecycle, you can just post, eg, 'Bag of kids' clothes, suit boy aged 5-8, useable, not tatty, some Next. To be collected from doorstep...' . Don't put your phone number on the Freecycle list as you may find yourself deluged with calls! After it's spoken for, don't forget to post saying item gone subject to collection (STC). I always make sure I go to the local Freecycle Yahoogroups website and delete my 'offered' post after the item has gone, too, otherwise lots of people seem to email asking for it without reading through the messages first to check whether it's gone or not.
I have been very careful about giving things to charity shops since I found out how much they throw out. A charity shop manager wrote to a national paper complaining that people were selling on eBay and she needed more donations as, apparently, only 20% of what she received was saleable - so the rest was binned. Now what one charity shop manager considers rubbish, another might think is great - much of it down to personal taste. Knew one round here who just binned anything from Primark, regardless of condition. The shop got around £1 for a black bag full of fabric and it's less effort than putting the stuff out in the shop. We are not in a posh area btw, so there are lots of people who would like the opportunity to buy cheap clothes. I really resent the fact that people have spent time and energy washing their donations and delivering them to the shop, then they are just thrown in the bin - so actually I think it's fairer to do what happened to the OP and be honest and say the shop can't use them. I only take stuff to charity shops now if I am pretty sure they will use it, and if I have tatty clothes I put them in a separate bag and ask the shop to give them straight to the rag man.
I have gone rummaging in charity shop bins (careful not to leave a mess!) on many an occasion, and found plenty of perfect stuff which was just thrown into landfill. Just to be clear, this is stuff in the commercial large wheelie bins that the shop has put out for the dustmen, not the unsorted donations that people leave on the doorstep, so there is no question that the shop was throwing this stuff out. Sometimes I've gone in and asked if it was OK to take, or put a donation in the tin, but am reluctant to do that now after a mouthful of abuse from one old moo. Finds from our local bins included a 3-person tent, neatly packed up & fine, which I used for a family camping trip and has since been passed on to another family. Sometimes I have taken stuff out of the bins and put it on Freecycle, because I can't stand the waste of perfectly useable stuff. Found things like working LeapPads and plenty of toys in good condition, which went down really well on Freecycle. It varies from shop to shop. There are about 5 or 6 charity shops on our high street and depending on storage and selling space, as well as the preferences of the sorters, some are more choosy than others.
Another option to support your favourite charity, and one they'd love, is to sell your unwanted stuff on eBay or at a car boot sale etc, then donate the money to them - but that does take a major time investment.
Is there a parent/toddler group or playgroup where you could put a card on the wall? Or does the school have a newsletter or website which takes adverts?