The Gaelic type is not an alphabet. It simply uses a diracritic sign (the dot over a consonant).
In my opinion, the language would need a proper alphabet in order to make clearer the difference between broad and slender consonants; using silent vowels makes things too complicated for learners.
Because it has more sounds than the Latin alphabet has letters (for example, broad and slender versions of most consonants), so they had to come up with tricks to indicate those extra sounds.
Also because Irish like to show the base form of a word in mutated (lenited/eclipsed) forms, so they add letters rather than changing the initial letter the way Welsh (for example) does - where penn "head" would go to benn when lenited rather than to bpenn, etc, or how Cornish writes "an venyn vyghan" for "the small woman" which would be "an bhenyn bhyghan" in Irish style (the base words are "benyn" and "byghan" and they are lenited much as in Irish).
When you use feminine singular nouns with the definite article an, you add a lenition to the noun (if possible).
- Maidin mhaith = (a) good morning
- An mhaidin mhaith = the good morning
- Bean mhaith = (a) good woman
- An bhean mhaith = the good woman
Bhean mhaith would not be a grammatical phrase on its own.