No. The law doesn't say talks on sexual orientation or gender identity are prohibited in class but that it should only be taught or discussed as per the academic standards set by the state for these subjects. This is clearer if we omit the phrase "in kindergarten and grade 3" in the proposed law:
Classroom instruction by school personnel or third parties on sexual orientation or gender identity may not occur in a manner that is not age-appropriate or developmentally appropriate for students in accordance with state standards.
Of course, this does mean that the state can very well decide that kids below grade 3 are too young to learn about sexual orientation and gender identity, and accordingly set standards that prevent teachers from discussing these subjects in class.
(As pointed out in the comments, the education board of the state is indeed involved in a public tussle with College Board to modify its content in AP Psychology to omit the subject of sexual orientation and gender identity for grade 4 and above - Florida effectively bans AP Psychology for gender, sex content: College Board).
Nevertheless, even in such a situation, it doesn't mean that teachers will be expected to tip-toe around sexual orientation or gender identity when teaching or discussing about relationships.
For example, consider this scenario - a teacher is telling a fictional tale about a kid named John (who has heterosexual parents) and mentions the mom and dad. A student in the class having homosexual parents may get confused with the reference of a female mom and male dad. He may interrupt the teacher and say, "Sir, how come John has a mom who is girl and a dad who is a boy when my parents are both girls?". The teacher may simply say, "Oh, some kids have mom and dad who are both either boys or even girls, like yours, and some have a girl mom and a boy dad, like John in our story. You will learn more about relationships in grade 4."
Sure, some curious kid will talk about this at home. And a homophobic parent may become upset at hearing this. They may file a complaint to harass the teacher. But in a fair court, the complaint will be very likely dismissed because the teacher acted appropriately and to the letter of the law. (Of course, if the matter has reached the court, that would imply that the school administration did not treat the complaint fairly. Sadly, such laws can certainly be used to harass teachers, and that could definitely be one of the political intent behind it.)
And of course, just like the homophobic parents, there may be pedantic people who may file a complaint against the teacher (or the school) for teaching a story with a straight couple. ;) Either way, Florida teachers are going to have a tough time ....