So the problem is: Views exposed filters don't have much configuration for the UI layer. This is where Better Exposed Filters comes in. Fantastic. Checkboxes, links, radios. The business.

Its a bit of a head twister to get Better Exposed working. I'm not clear on the logic of the placement of the settings which is why it may have taken me a little longer than it should've (the settings in D7 are in Advanced > Exposed form style and its settings panel).

Another hurdle is that Views doesn't provide a select list UI for Entity References. This means that (in my situation) I couldn't initially use Better Exposed (as I was filtering on an Entity Reference). I could have reworked my entities as a taxonomy but instead I came across this issue.

Which mentions that there are limited lists available in the current version of Views. Where? As it mentions in the comments there is an enabling checkbox at the end of the entity reference field settings edit page 'Enable Render Views filters as select list'.

After enabling that (and removing and readding my filter) I could set my Advanced > Exposed form style to Better Exposed Filters and in settings enable Autosubmit, the hiding of the submit button and display the filter as radio buttons.

A remaining problem is that the links formatter from Better Exposed doesn't like AJAX. There are patches out--none of which worked for me. The CSS hack on top of radio buttons (plus a little javascript) in one of the final comments is a winner however.