Re: baby-led weaning snafu
You're over thinking. It's definitely not failing at BLW to offer purees sometimes. If that's what works at daycare, do that. I actually might try send more purees and ask them not to give her cereal (since it's basically just filler and there's little nutritional value to it).
I think you're being too hard on yourself. It sounds like your BLW experience has been going pretty typically. They don't eat much to begin with, it's a crapshoot what they like and don't like. Just keep trying different things and you'll hit on a routine. Don't stress about it.
As for the vomiting, my guess is that she got those tiny balls from the broccoli stuck in her throat and it made her gag.
ETA:One tip I'd give you is don't feel like you have to give tiny pieces. It's actually better to give big pieces of food with 'handles' so they can hold on to it and suck on it without the whole thing going down their throat and choking them. So brocolli is fine, but give her a big enough piece that she can hold it herself and decide for herself how much of it she wants to put in her mouth. Cut the pear into long french fry sized pieces, split the banana in thirds lengthwise and let her have a whole long piece, cut a sweet potato (again) into long french fry shaped pieces and bake them and let her hold and suck on one. Things like that.
“We are not put on earth for ourselves, but are placed here for each other. If you are there always for others, then in time of need, someone will be there for you.”