Thanks for your response, Chris.
I've been working with your first few suggestions - sticking with what I have. I hadn't considered implementing that 'Yelp-style' click. That sounds like it could be a good option for me. I'll search around to see if there is some related code up on a github (unless you know of an example?). I've used marker clusters before, because I don't really need to see every distinct point at a wide zoom, but there doesn't seem to be much of an impact on speed in any case.
I have a little bit of experience with setting up GeoServer & PostGIS, but my 'coding' skills are limited primarily to R. So I've been trying to squeeze as much juice out of the Shiny/Leaflet fruit as I can, so to speak.
The more people I share this little app with, the more the excitement around it grows. Consequently, the more concerned I get that I'm going to be asked to do something beyond the limitations of this framework - hence my initial question. In truth, the app isn't really for analysis or even for concurrent users (I do the geospatial analysis in R itself, without needing to visualize it). It's more of way to market capabilities that look a little less sexy as lines of code :)