History

The reason for the set mix as it exists is that, originally, the dice available were a set of platonic solids, sold by an educational company and repurposed by TSR. Namely, a tetrahedron (d4), cube (square hexahedron, d6), equilateral octohedron (d8), dodecahedron (d12), icosahedron (d20). This was a "platonic solids" set.

D20's were routinely read as d10's in wargaming, and used for generating percentiles, which made use of the d20 as a d10 or pair as a d100 a standard practice, going back as early as 1972.

A 1938 dodecahedron with 2 rounded faces opposite each other was available in 1938 for stock simulations. ➀ It wasn't used by TSR, but shows the probable origins of the pentagonal dipyramid we now think of as the d10.

Names

As to names - the earliest modern set (with dipyramid d10) I can readily find documented by name is the "Dragon Dice" set from 1981 (as evidenced on the packaging copyright notice), as photographed at Dice Collector. ➁

I can, however, cite TSR "catalogue" from a ©1979 TSR product - Swords & Spells lists the older set as "Multi-Sided Dice Set" in the catalogue extract on the back flyleaf. ➂ It reads:

Multi-Sided Dice Sets — Each set contains one 20-, 12-, 8-, 6-, and 4-sided die

The same title and text was used in the 1975 product list in Strategic Review Vol 1. Issue 3 (1975). ➃

So:
Multi-Sided Dice Set - the original d20 d12 d8 d6 d4 platonic set.
Dragon Dice - the "mud dice" set with d10's.

➀ Dice Collector - Mason & Co Stock Exchange Dice - www.dicecollector.com/MINT39_MASON_&_CO_STOCK_EXCHANGE_DICE.jpg

➁ Dice Collector - TSR - www.dicecollector.com/THE_DICE_THEME_TSR.html

➂ Gygax, [E.] Gary, Swords & Spells, 6th printing, Tactical Studies Rules, 1979. Original copyright 1976.

➃ Tactical Studies Rules, Strategic Review, Vol. 1, No. 3, Autumn 1975, Ed. [E.] Gary Gygax. TSR advert on page 8. From the Dragon Archive CD.

At the moment I can only see two solutions to this particular problem. The unfortunate thing is that both solutions offer their own challenges and difficulties as well. I agree with you that rolling a bunch of dice and adding it all together is tedious- let's see what we can do about that!

Let the Robots take care of it

For large sums of dice getting rolled together, a great way to speed up the process is to use a Dice Roller program that can be found through a quick Google search (I'm at my office right now and can't link you, sorry.) You type in how many dice you want to roll, select the value, and the computer instantaneously gives you the sum.

You can either use truly random dice generators like Random.org's, which are based off atmospheric data. Other generators will be pseudorandom number generators which run off a mathematical algorithm. The difference between either doesn't really matter unless you're doing cryptography or quantum mechanics, so either is just as good as the real dice for people playing RPG at a table.

Shortcuts

You could also, when there is a ridiculously large amount of dice to be rolled, use average value instead. So 40d6 becomes 140 damage (3.5 x 40). No rolling, just flat damage.

The Problem

People like rolling dice. It is the part of the game that is most "tactile" and therefore is very satisfying to many people who enjoy the hobby. Clicking a button on the computer can sort of emulate the feeling, but just doing average value cheapens the experience. Not to mention that some players may balk at the idea of not getting the chance to do more than average damage with their awesome spell/ability.