It will be difficult to find an authoritative source for the reason as it is almost certainly complex and very ancient. However one possible answer is that it is related to the old habit of counting in twenties or "scores" (think of the start of the Gettysburg Address "Fourscore and seven years ago..." and the biblical quote "The days of our years are three score years and ten...") . The reason for special words for 11 and 12 could also be related to counting in twelves or "dozens".
English is not alone in having special words for numbers between 10 and 20, German has 'elf', 'zwolf', 'dreizig','vierzig 'funfzig' and so on and French has 'onze', 'douze', 'treize', 'quatorze', 'quinze' and 'seize' although it reverts to 'dix-sept','dix-huit', and 'dix- neuf' for the last three.
Interestingly French has echoes of the old 'counting in scores' habit since the word for 80 is 'quatre-vingts' or 'four-twenties' which is an almost exact translation of 'fourscore'. The French for 90 is 'quatre-vingt-dix' or 'fourscore and ten' (note the lack of an s on 'vingt') and the French for 70 is 'soixante-dix' which translates as 'sixty-ten'. The numbers from 71 to 79 and 91 to 99 incorporate the numbers for 11 up to 19 so that 61 is 'soixante-onze' and 89 is 'quatre-vingt-dix-neuf'
There are also traditional counting systems in many British dialects which were specifically used for counting sheep and these were based on twenty. In fact they have no words for numbers above twenty and each time they reached twenty a notch (a 'score' perhaps) was made in a stick. A flock of 93 three sheep therefore would have been counted as 'four notches and thirteen', effectively 'fourscore and thirteen' or 'quatre-vingt-treize'.
The Lincolnshire dialect sheep-counting numbers up to 10 were
Yan, Tan, Tethera, Pethera, Pimp, Sethera, Methera, Hovera, Covera, Dik
and the ones from 11 to 20 were
Yan-a-dik, Tan-a-dik, Tethera-dik, Pethera-dik, Bumfits, Yan-a-bumfits, Tan-a-bumfits, Tethera-bumfits, Pethera-bumfits, Figgits
As noted above the shepherd cut a notch in a stick every time he reached Figgits. It is also interesting that there is a special word for 15 which fits no pattern.
There were regional differences in these number systems but they all counted up to twenty and no further. This website gives some of them in a table.
In summary the origins of the names for numbers are many and various and the names of the numbers from 11 to 19 are so well established that there is really no need to change them. This website which discusses French numbers says that the French-speaking parts of Switzerland and Belgium have 'regularised' the numbers 70, 80 and 90 to 'septante', 'huitante' and 'nonante' but the rest of the French-speaking world seems to stick with the 'counting in scores' system. The Swiss and Belgian change simplifies the French number system greatly but moving from "eleven, twelve, thirteen" and so on to "ten-one, ten-two, ten-three" etc would cause more problems than it would solve. It probably won't happen.