I'm interested to know if FamilyBase has the functionality for us to read texts - as we feel needed - that are on our kids' phones, if the kids are covered under FamilyBase. If not, is there another way via Verizon to do so?
No it doesn't. In some states that is even illegal if you didn't tell you kids you wanted to do that.
You can setup messages+(Verizon thing) which I think works the same way. Just keep in mind check you local laws so you don't shoot yourself in the foot. There are third party apps that do it as well.
Check your local laws. Like here if I give my kids a phone with a set of rules and they agree with them then I can enforce those rules, but if they don't agree with them I don't have to give them the phone kind of deal like an EULA.