I visited Hanlin from Mandalay in December 2013, just using a private taxi driver. I can't give any specific advice regarding any agencies, as I didn't use one, nor can I give any advice regarding the practicality of Shwebo, as I didn't go through there.
The taxi driver I spoke with had good English, and seemed a bit confused as to why I wanted to go to Hanlin. He knew of it, but had never been (and actually asked if he could bring his friend too, so it was obviously a bit of a curiosity). The main road from Mandalay to Shwebo was decent enough, but once the turn off was required, the quality of road deteriorated quite quickly. The taxi managed well enough, but it was a bit slow going. I think it was about 2 hours from Mandalay to the site? I assume a guided trip would have a better vehicle and thus be much faster over the rutted dirt roads. A motorcycle driver based in Shwebo would presumably know the area better and be more mobile, though that has the added hassle of getting there from Mandalay in the first place.
At the actual site, you go to the 'museum'/ticket office. When I was there it was under rapid construction, so it is probably properly finished by now. After paying, we were supplied with a guide on motorcycle, who the taxi was instructed to follow. The small map supplied on the ticket listed a large number of individual sites, but we apparently had no say in which ones of these we were to see: our job was to follow the leader. I assume they took us to the most interesting parts, though.
The guide did not speak any English, but was obviously enthusiastic about the topic, so our taxi driver translated for us. This is probably the deciding factor as to whether you would want to find a 'proper' travel agency guide, as your understanding of the archaeological elements rests entirely on the translation of your transport provider. Our taxi driver was enthused at first, but it was obvious we were getting a very simplified explanation due to practical language issues. By the end, a 90 second speech by the guide was sometimes translated as 'This is also a wall', or similar. Given that the remains are often little more than foundations and rubble, a proper explanation of what you're actually seeing is pretty important, so if you use your own driver, make sure you're happy with their English skills and dedication to translation.
It was a very regimented experience of following the guide to the next station even as a 'private' visit, so I don't expect hiring your own driver would allow an increased level of freedom compared to using an agency for a day trip.
Also, if you do go into the human remains sites, (I'm not actually sure how these relate to the OUV of the site, but they were part of the tour anyway), maybe consider politely refusing if the guide suggests you go into the actual pit. As I mentioned in my review, my guide stepped on a leg bone and shattered a bit of it, which was pretty shocking.
Sorry if that was a bit rambling, but hopefully it was useful.