The Lucas fuses (and the fuse ratings given in the Triumph manuals) are rated by the current that makes them blow "instantly". But virtually all fuses today are rated by the current they will carry "forever". The difference is roughly 2:1, so a "35 amp" Lucas fuses is roughly the equivalent of a modern 20 amp fuse!
Glenn Merrill and one of the other fine folks from ISOA (whose name I forget offhand, sorry!) put on a most convincing demonstration at VTR 2005. Using a short section of original harness wire, they installed a Lucas fuse and loaded it until it blew. Then they installed an AGC fuse (AGC is a common US fuse style, very similar in size and appearance to the original Lucas fuses) with the same rating and the same load. It took a few seconds, but in fairly short order the wire started visibly leaking smoke, and moments later the insulation started falling off!
So the moral is that you should either buy reproduction Lucas fuses (usually available from the usual suspects, Moss, TRF et al); or if you want to buy fuses at FLAPS, use half the rated current. The choice is yours, of course. The AGC fuses are very slightly different physical sizes than the Lucas fuses, but they fit well in all of the fuse holders I have. Seems like someone said they wouldn't fit in his TR8 (but I don't have anything that new).
The later cars with the tail light fuse (like my TS39781LO, which also had a switch rather than rheostat) did have the instrument lights powered through the fuse. I believe this was considered a safety feature, as otherwise there is no visible indication that your tail light fuse has blown. Of course that doesn't help if you have them switched off
My personal take on the fuse situation is that these were inexpensive cars (at the time) and electrical components were expensive. Standard-Triumph was more or less forced to buy electrical components from Lucas, meaning Lucas had little incentive to keep costs down. And part of the Tr3s popularity was that it was cheaper than other sports cars at the time.
But, there is also a safety aspect. Fuses back then were not overly reliable, and sometimes failed for no apparent reason. I had added fuses for the headlights on TS39781LO, and one night while driving down a mountain (rather faster than I should have been driving), the headlights simply went out! Fortunately it was on a relatively straight section of road, and after a momentary panic, I figured out to punch the dimmer switch and finished driving into town with the high beams on. But it could have been much worse! I couldn't find anything wrong, so next morning I changed the fuse and we continued our trip. The bad fuse looked funny, not obviously blown but there was a silver blob visible at one end of the glass tube.
When it happened again a few months later (this time just driving to work on a gloomy day), I finally figured out that the fuse clips had a tiny bit of corrosion on them, not enough to see, but enough to create some small resistance. Over time, the resistance created enough heat to eventually melt the solder in the fuse caps! I polished them up with a bit of Scotch-brite and the problem never happened again.
One of the old manuals discusses this in conjunction with the overdrive, warning that a fuse failure could cause a sudden shift back to direct drive and overspeed the engine. So apparently at least the folks at Laycock didn't trust fuses.
In case you don't have one, here is a factory diagram showing the tail light fuse with the dash switch. This is from "Practical Hints" (which is effectively the owner's manual from Triumph) 4th edition.
Sorry, I don't have the 4th edition completely scanned yet, but here is a link to the 3rd edition
https://drive.google.com/open?id=1Ma-NUXwwAwds9DxYYQgtVFgkclrXyETu