Hi, we chose a voluntary agency so I can only tell you about our experience but having spoken to a lot of other adopters during the process I don't think there is a 'best' route - just what is best for you.
We started the process in October and are currently waiting for matching panel for our 2 little ones that will hopefully come home in October this year - so the process has been pretty fast for us.
We met with our local authority and 2 VA's that operate in our area -
LA was a nightmare to deal with, never called when they said, promised us information/invites to information events that never materialised, took weeks to get back to us about anything. We researched them online and found lots of people saying the same and their journeys taking years because of it, so we decided they weren't the right fit for us.
VA1 was very slick and professional, sent us really glossy information and seemed impressive, but the social workers we spoke to didn't feel welcoming and they made the process sound very intimidating.
We picked VA2 in the end, they are a smaller, local agency and immediately we felt incredibly welcome and supported, the social workers have all been amazing and we are very happy with the decision.
The main 'benefit' of an LA is often said to be that they are responsible for the children, so placements can be made sooner when the child is younger. So if this is important to you, you would probably be best suited to an LA, regardless of how difficult they can be to work with.
For us, we knew we wanted a sibling group and every agency we spoke to said they were 'desperate' for adopters for siblings, so we didn't feel like we needed to deal with the LA to get what we wanted.
In our time family finding (it took us about 3 months after approval to find our right match) we've been approached about 80+ sibling groups with children ranging from unborn - age 6 (which was the maximum age that we set), so I can honestly say being with a VA has not limited our options in any way - but if you are hoping to adopt a single child under 1 you may have a very different experience with a VA as those placements are likely to be few and far between.
My advice would be to speak to them all and see what feels right - there isn't a 'best route'.