Hi Ben, briefly took another look this morning and I'm sorry for the hasty rec because I forgot Leaflet doesn't support vector tiles as easily as MapLibre/Mapbox.
If the bottle-neck is the DB and you haven't enabled the PostGIS extension, I'd look into that as the other commenter mentioned. If it's your server, then yes look into pre-generating all the tiles and hosting them elsewhere, but this might then mean switching over to MapLibre/Mapbox (which is what most companies end up using because of Leaflet's limitations).
For filtering on the frontend, I'd look into Expressions so you're not regenerating the GeoJSON collection every time (this causes a significant delay and flicker).
You could also take a look at PostGIS to do geospatial queries (if needed). And in the past I’ve used node-mapnik to render vector/image map tiles. You can indeed host and cache them effectively on S3.
