As someone who has been to GayDays for the last several years, my advice is to go and have a great time with your family, avoid the MK on Saturday June 7, it will be crowded. Pleasure Island is not an area I'd consider "kid friendly" after about 8pm, no matter what time of year, nightclubs, drinking and kids don't mix.
If you avoid the "park of the day" you'll probably NOT notice one thing unusual. Even the "designated" parks are generally OK for families with children.
Of course, only you can make these decisions for your children, but I take my 78 year old Mom, who is in a wheelchair, and have yet to see anything I'd consider "over the top".
Remember that many of us will also be traveling with our own parents/children/families and share many or the "public display" values as every parent/family.
Saturday is, of course, the busiest, most crowded day. The other parks, designated or not, have light to medium crowds. Personally, I don't find an "empty" park nearly as enjoyable as a somewhat busy park...guests add to the overall experience.
Schedule is:
Thursday - AK
Friday - Disney/MGM
Saturday - MK
SUnday - EPCOT