Blocks need to be put on the page, or someone to add them as a 'sticky' block across the site.
Adding the functionality to a local plugin should remove that requirement as it would always be available in the same way as something like adminer. That said, it may be that you want this to float over the current page (adminer does open its own in the background) in which case maybe a block is better.
To be perfectly honest, where you have written blocks before, I have written quite a few themes, so I would have been tempted to build this functionality into a theme - but again, if your institution needs to use multiple themes, that may not be appropriate either
As far as I can tell, the need to add the block to the page(s) is the only way it would affect behaviour/usability that I can think of and that could be an advantage in terms of controlling where it appears (or doesn't). But the suggestion really came from having just been using adminer this morning and knowing that that has a kind of 'floating UI' and is installed as a local plugin
R