With thanks to Leaflet.js forum members Matt and Simon.

I was encouraged to try again to obtain up-to-date LA boundary data from the OS. This time I obtained a set of shape files. The relevant shape file was district_borough_unitary_region.shx. I opened that in QGIS, removed the non-Scottish areas and saved the result as Scotland.qgs. I then selected and saved each LA in turn as a geoJSON vector file (with CRS = WGS84/EPSG4326 as before). So this got me up-to-date accurate LA boundaries. But these are quite large files: averaging 2MB each. (Of course Highland was much bigger than Clackmannashire, for example.)

To obtain simplified files, I selected each LA in turn, then did Vector > Geoprocessing tools > Simplify (options: ‘use only selected features’, tolerance = 50, ‘save to new file’, ‘add to canvas’) so that I had a patchwork of separate LAs:

I then selected each simplified LA in turn and saved them as geoJSON files. The resulting files are about a twentieth of the size of the unsimplified files. Yet zooming right in shows an acceptable (to me) fit to the unsimplified boundaries and coastlines. Yeehah.

We have layers

And now the LA group is off by default, while the locater and layer controls are expanded by default: