Unfortunately the specific answer to your question is "yes the law might forbid children from being out and about without supervision, but it depends on the state". I believe the section of law most applicable to this question is "child neglect", and for the most part the definitions of child neglect are left to the states. There is a very wide variety in what is considered "neglect" in various states.
There are often specific laws involving "leaving children alone at home", but they usually don't have much to say about leaving children unattended outdoors. For example, Maryland's 5-801 states that children under 8 cannot be left alone (confined) under the supervision of anyone younger than 13. Note that although this statute does not technically apply to children left to "run free", it is certainly not too hard to imagine a state judge interpreting it as being "in the spirit of the law".
To stick with Maryland again, this is the definition of neglect from 5-701
(s) “Neglect” means the leaving of a child unattended or other
failure to give proper care and attention to a child by any parent or
other person who has permanent or temporary care or custody or
responsibility for supervision of the child under circumstances that
(1) that the child’s health or welfare is harmed or placed
at substantial risk of harm;
This definition is a bit non-specific, which could potentially cause trouble.
The department of health and human services provides a rather nice summary of state laws as they apply to neglect (and many other crimes against children): https://www.childwelfare.gov/systemwide/laws_policies/statutes/define.pdf . They really do run the gamut as a judge could potentially apply them to 'free range children':
Neglect Citation: Alaska Stat. § 47.17.290 ‘Neglect’ means the
failure of the person responsible for the child’s welfare to provide
the child necessary food, care, clothing, shelter, or medical
It would be a bit of stretch to get any sort of trouble out of this.
Neglect Citation: Rev. Stat. § 8-201
‘Neglect’ or ‘neglected’ means:
• The inability or unwillingness of a parent, guardian, or custodian
of a child to provide that child with supervision, food, clothing,
shelter, or medical care if that inability or unwillingness causes
unreasonable risk of harm to the child’s health or welfare
(bolded the parts that could potentially be troublesome)
Neglect Citation: Ann. Stat. § 626.556, Subd. 2
Failure to provide necessary and appropriate supervision or child care
arrangements for a child after considering such factors as the
child’s age, mental ability, physical condition, length of absence, or
environment, when the child is unable to care for his or her own
basic needs or safety, or the basic needs or safety of another child
in their care
Mostly someone would be working with the definition of 'safety' here, but for the most part this is pretty reasonable.
Neglect Citation: Welf. & Inst. Code § 300
A child may be considered dependent when:
• The child has suffered, or
there is a substantial risk that the child will suffer, serious
physical harm or illness as a result of:
» The failure or inability
of the parent or guardian to adequately supervise or protect the child
This... could be stretched to something troublesome. Similar to Maryland in some ways.
Since you specifically mentioned Colorado, I'll point out that the definition of neglect there is maddeningly flexible.
Neglect Citation: Rev. Stat. §§ 19-1-103; 19-3-102
The term ‘child abuse or neglect’ includes any case in which a child
is in need of services because the child’s parent has failed to
provide adequate food, clothing, shelter, medical care, or supervision
that a prudent parent would take. A child is ‘neglected’ or
• The child lacks proper parental care through the actions or
omissions of the parent, guardian, or legal custodian.
• The child’s
environment is injurious to his or her welfare.
• The parent,
guardian, or legal custodian fails or refuses to provide the child
with proper or necessary subsistence, education, medical care, or any
other necessary care.
(I have edited the quotes from the above-linked pdf only to remove definitions irrelevant to the discussion... but please note that all of them have been edited, and should not be represented as authoritative summaries of the referenced statutes)