The vowels in the English language: bat, bait, bet, beet, bit, bite, bot, boat, but, butte, bout, boil, book, the 'a' in about (/ə/), baht. That's only in my dialect, and not counting r-colored vowels (which are not r-colored in British dialects!).
This list was constructed by building short/long vowel pairs, and then tacking on the weirder vowels afterwards. As you'll notice from the inclusion of the last entry, there is a major dialect issue here. There are four sounds (/æ/, /ɒ/, /ɑ/, /ɔ/) that are usually distributed into three sounds in major dialects, and these arise from short a and short o in written orthography... but which words send their vowels to the different targets varies greatly depending on the dialect. If you pick it according to any one dialect, you're going to get some poor correlation in another dialect.
The other obvious issue is that adding new characters is actually difficult in the modern age. Of the sounds I've dropped in above, I can only type in ə and æ without resorting to copy-paste, and that's only because those are accessible with the Compose key. Telling people that you have to use 10 keys not present on their keyboards to write their language is going to be a hard sell. Again, note that English dropped þ as a letter primarily because it wasn't possible to print with printing presses imported from Germany.
This list was constructed by building short/long vowel pairs, and then tacking on the weirder vowels afterwards. As you'll notice from the inclusion of the last entry, there is a major dialect issue here. There are four sounds (/æ/, /ɒ/, /ɑ/, /ɔ/) that are usually distributed into three sounds in major dialects, and these arise from short a and short o in written orthography... but which words send their vowels to the different targets varies greatly depending on the dialect. If you pick it according to any one dialect, you're going to get some poor correlation in another dialect.
The other obvious issue is that adding new characters is actually difficult in the modern age. Of the sounds I've dropped in above, I can only type in ə and æ without resorting to copy-paste, and that's only because those are accessible with the Compose key. Telling people that you have to use 10 keys not present on their keyboards to write their language is going to be a hard sell. Again, note that English dropped þ as a letter primarily because it wasn't possible to print with printing presses imported from Germany.