This gets confusing because the state can play two separate roles in adoptions, and people conflate the two.
Two things have to happen for an adoption to take place: the parents and children have to be matched up, and the adoption has to be approved.
The former can be done by the state, by private agencies, or between individuals. When the state does it, it is looking to place children whose parental rights have been terminated. These are never (AFAIK: there may be the occasional exception) newborns simply because even if someone is a serial killer in prison with no interest in hindering the process (but not actively expediting it), it still takes about six months to go through. These kids can be placed in the homes of adoptive parents before this (foster-adopt), but nothing can be made permanent until the parent loses their rights. The state is also the group most likely to be placing older children, because older children who are available for adoption are virtually always the result of legal loss of custody: not many people are actively looking for adoptive parents for their three year old. In all these cases, the terminating of parental rights can take a long, long time–months to years. It’s not something the state does lightly. These kids may also be placed with potential parents in the meantime (foster-adopt, again). As mentioned above, the adoptive parents are “pre-screened”, so there isn’t much waiting on that end: you don’t even start the process with a potential parent(s) until you are sure they are ok. This option is very low cost, and can be very quick if you aren’t “picky”: parents can place limitations on the sorts of children they will accept (like “no sibling groups” “only newborns” “no history of sexual abuse”) and obviously the pickier you are, the longer it might be before you get matched up.
When agencies get involved, it is because someone has voluntarily decided to give up a child and is looking for potential parents. They go to the agency, which has compiled dossiers on various parents who are looking to adopt, and the bioparents can basically look through them to find a family they want to place their child with. Details such as open/closed and such can be worked through there. The agency usually has some sort of screening process as well. Again, the parents “available” have been pre-screened and each agency will have its own standards. Generally in these cases the baby goes home from the hospital with the parents, but there is generally some sort of waiting period before the paper work is filed. This is much more expensive, and the time really can vary before you are matched up because the bio-parents have a lot of control (which is not crazy: if you are giving up your child, you’re going to go to the agency that allows you to screen). Some people are selected quite quickly: some people wait years.
Last is totally private adoption, where it is arranged between two individuals. A lot of these cases are inter-family, or between people already known to each other. Each of these is different.
In all three cases, the state enters into it again when it comes to approving/performing the adoption. Obviously, the stat acting as an adoption agency doesn’t place kids in home that the state as adoption-approver won’t allow. In terms of private agency adoptions, parents can be approved by an agency but not by the state or by the state but not the agency. I think that in most states, the investigation/approval process of reputable agencies suffices, but I am not sure about that. In terms of totally private adoption, the state certainly has the right to reject an adoption application even if all parties agree to it.
These two roles of the state: placing children (in some cases) and approving adoptions (in all cases) is often conflated and is part of the reason people get confused in debates about gay/older people/atheist/single-parent/whatever adoption. State adoption agencies may have policies to not place kids with parents that they would still approve in cases of private adoption.
TL:DR: There are two processes here: the placement and the legal adoption. The timeline on the first varies widely and the timeline on the second can’t even start until the first is in place.