No worries . My son was the same way . In fact it's the texture not so much the chewing they take issue with . Imagine if you spent a year drinking and now someone plops something to chew on a plate ..ick .
I used the following to intro food to my son and it seemed to help .
Toast with formula , Sounds Yummy doesn't it ? Seriously the wet toast with the familiar taste gets them used to thicker textures.
Pears ..way over cooked . They are sweet and dont require much chewing.
Carrots ...way way over cooked ( see pears)
Broccoli ( this is a weird texture and it helps them really learn to chew and my son LOVES the taste , maybe yours will too.
My best tip , don't mix textures. My son will still spit out chunks in the food he eats and then pick them up and eat them seperately ( he's almost 20 months) .
As for the cheerios , try gerber puffs. They are far easier to chew and tastier than cheerios . He will probably do better with those for now.
AND about the Day care provider. No peanut butter till 3 ( according to the American acad. of Pediatricians) Is she trying to give these kids Peanut Allergies? I Refused to serve my niece ( who I used to watch) any peanut or nut product for fear that she'd go into shock while I was watching her . My husband knows the rules as does everyone else regarding my kids and nuts, shell fish and certain fruits which my husband is allergic to.
Hope she's adept with an epi pen