I'll try to answer all those questions.
First, there are at least three types of road associations. An "Informal" road association would be where neighbors just voluntarily pitch in towards the cost of road maintenance. They don't have to file with anyone, or follow any particular rules, and they have no enforcement powers.
The second type is the "Non-profit Corporation Road Association." It has to file with the State as a non-profit corporation, and follow the rules for non-profit corporations of any type, including paying an annual fee to the State. It has no specific enforcement powers, although it does have the option of filing a complaint in small claims court to try to collect fees from a non-payer.
The third type is a Statutory Road Association, which is set up under very specific rules set forth in 23 MRS 3101 - 3104. Here's a link: https://legislature.maine.gov/legis/statutes/23/title23sec3101.html You can scroll through the applicable statutes by clicking on the arrow at the top right hand corner of that page.
In order to be a Statutory Road Association, three or more owners of property benefited by the road have to get together and apply to a Notary to call the initial meeting, and they must send notice of the meeting to every owner of property benefited by the road. It is only by carefully following the process detailed by the statutes that an association becomes a "statutory" road association, and dues collected are to be used ONLY for road maintenance and related expenses. (Common land is not covered.)
It is possible for a road association to satisfy the requirements to become BOTH a non-profit corporation and a statutory road association at the same time, and many do that in order to get the benefits of both. For example, as you state, a non-profit corporation may also care for common properties, which a Statutory Road Association cannot do. But a Statutory Road Association has the power to file a "Notice of Claim" (basically a lien) to aid in the collection of unpaid dues, and a non-profit corporation does not have that power.
So to answer your questions specifically, you said, "I gather that the 'teeth' in collecting fees for private roads is granted to road associations." To be clear, it is granted to STATUTORY Road Associations, not to non-profit corporations.
You asked, "And a non profit who has the duties of both the road association and common facilities, MAY fall under these statutes or not?" A non-profit corporation does not fall under 23 MRS 3101-3104, but can ALSO follow the requirements to become a statutory road association for the purpose of maintaining the road (not the common facilities) and having the powers granted by those statutes.
You asked, "only an entity that operates under the 'road association statutes' has the authority to put a lien on a property for non payment; so if there are common facilities beyond road maintenance, then the portion of the fees relating to common facilities may not fall under road association statutes?" That is correct. Fees for the common facilities cannot be charged by a Statutory Road Association, so it cannot use its powers to collect them.
You asked, "If there is a non for profit, that is not functioning under the road statutes, then they do not have the authority to place a lien on properties?" They do not have the authority to file a Notice of Claim. They do have the power to file a claim in Small Claims Court, but that involves actually going to Court. A Notice of Claim is simply filed with the Registry of Deeds, then the Notice just sits there until it either expires after 18 months, or is renewed before it expires. If the land owner tries to sell the land or obtain a loan, a title search will bring up the claim and it will need to be settled to clear the title. So it doesn't always bring speedy payment, but it is likely to get paid eventually.
You asked, "Is it possible for a non for profit that maintains roads and common facilities to function outside of the road association statutes?" Yes, it can operate purely as a non-profit corporation, but 23 MRS 3101-3104 will not apply. It will not have to go through the process of having a Notary call the first meeting, etc, and will not be bound by the limits imposed by those statutes, (restrictions on where paving is allowed, for example.) But it also will not have the advantages afforded by the statutes such as the ability to file a Notice of Claim, and the Immunity from Suit that is afforded under section 3101.
You asked, "Are there many 'non for profits' in the state that choose to function outside of the road association statutes?" Probably, although I can't tell you how many.