I always assumed that 'Tadhg' was the correct spelling of the personal male name and that 'Tadgh' was a flagrant misspelling, arising from an ignorance of Irish orthography. But is there any chance that 'Tadgh' is also correct, after all?

For instance, there are a small few words with lenited consonants where the lenition isn't indicated in the spelling e.g. 'taitneamh' (when you might expect 'taithneamh'), 'fidh'/'faidh' future-tense suffix (when you might expect 'fhidh'/'fhaidh'). I'm fairly sure there are more examples but I can't think of any off hand.

On the other hand, if 'Tadgh' (with implicit lenition of the 'd') was correct, this would only make sense in the context of Munster pronunciation, where final 'gh' is de-lenited to 'g'. But in Connacht and Ulster, final 'gh' is always silent (as far as I understand), which would yield a pronunciation approximating to English 'tie' - but I have never come across such a pronunciation.

Níl ' fhios agam, a Chaoilte, ach 'sé 'n rud is mó a chuireann orm ná fuaimiú na hainme seo. Tá canúintí éagsamhla ann, gan dabht, ach i nGaelainn Chorcaí is /tˠəigˠ/ adeirthar, cé go ndeireann a lán daoine gur mar a' gcéad shiolla de "Tiger" adeirthar é, rud nách ionann leis, nú nách ionann ins gach canúint Bhéarla ar a' laíod....

I’m pretty sure it’s only slender gh that gets delenited (hence tigh pronounced as if tig), but generally not broad gh. (though you do get /əɡ/ for the -adh ending in the past autonomous in parts of Cork)

As far as I know Tadhg is the correct spelling in modern Irish.
In older Irish you might see Tadg with a punc on the D.
I'd say that Tadgh is probably an English variant of it. I'm not sure really.

I'd pronounce it as the English word "thigh" (upper part of your leg) and then IG sound at the end.

A Djwebb, tá aineolas iomlán ann mar gheall ar bhfóineolaíocht na Gaelainne i measc an phobail i gcoitinne. 'Sé an bhunfhadhb ná nach féidir fuaimniú a fhoghlaim as leabhair amháin. Caithfear éisteacht le taifeadadh fuaime freisin, ar a laghad. Agus caithfear buneolas a bheith ann maidir le teoiric na fóineolaíochta, i mo thuairimse. Le ceart, thuigfí gur ar aon dul le h-uirlis cheoil a fhoghlaim é fuaimniú teanga a fhoghlaim. Mar sin ba cheart go mbeadh múineadh duine le duine ann mar ghné bhunúsach den mhúinteoireacht scoile chun aiseolas a ghabháil leis an bhfoghlaimeoir ó chainteoir líofa.
Níl mórán Gaelainne agamsa ach is féidir liom féin tuiscint gur mugadh magadh amach is amach é caighdeán múinteoireachta na Gaelainne ins na scoileanna, mar gheall ar an bhfuaimniú ar a laghad. Ach is ábhar ollmhór é seo...

Silmeth, I think you're right there actually. In Munster Irish, I think word-final broad dh is always silent, except in the case of the suffix denoting autonomous past tense (adh/íodh), as you point out. I think word-final broad dh is also always silent in other dialects, except that, in other dialects, if the preceding vowel is short, the vowel is liable to be altered to a long monophthong (or maybe to a diphthong), I think.

⦁ Examples of words ending with 'adh': geimhreadh, samhradh, conradh, ionradh (incursion, invasion); also verbal nouns e.g. bualadh, ceapadh, cleachtadh, etc.

⦁ Other examples ending with broad 'dh' (there seem to be very few examples where word-final broad 'dh' is not preceded by short 'a'): ádh; modh, aodh (inflammation); nódh (= nua); mudh (literary word, meaning 'destroy');

I've no idea how word-final broad gh should be pronounced in Munster or any other dialect, although there seem to be very few examples of such words.

⦁ Examples ending with broad 'gh': agh: (literary word, meaning 'cow', 'ox'); ágh (battle, contest); blogh (fragment (both noun and verb)); ógh (virgin), lógh (= luach); Lugh (the pagan god); súgh (=sú)

I've heard Irish broad 't' being described as being midway between English 't' and English 'th' i.e. tip of tongue is slightly forward in mouth but not as far forward as in English 'th'. I reckon this is a helpful way of looking at it from the point of view of a native English speaker.

But I find it interesting that you describe Irish broad ''t' as being pronounced like English 'th'. My understanding is that when the English language first came in, Irish speakers realised English 'th' as Irish broad 't' when speaking English, since Irish broad 't' was closer to English 'th' than Irish slender 't' was. Irish slender 't' was then used for English 't' (I think). THis practice might be deemed an older style of Hiberno-English.

Eventually, Irish people generally ended up mastering the English 't' sound but, due to some quirk unknown to me, Irish broad 't' generally stopped being used to represent English 'th' and instead English 't' started to be used, thus effecting a phoneme merger. This might be deemed a newer Hiberno-English.

I notice this where I come from, with some (probably a minority of) older more-conservative English speakers using Irish broad 't' to represent English 'th' (thus maintaining the phoneme distinction), but with everyone else pronouncing it as English 't'. Possibly the same conservative speakers use Irish slender 't' for English 't' but I can't say for sure off hand.

My guess is that this phenomonen of using Irish broad 't' for English 'th' is only observable in people who come from areas that were Irish speaking in the relatively recent past or that are currently Irish speaking. Hence my guess as to why you equate Irish broad 't' with English 'th'.

A Chaoilte, tá a lán Gaelainne agat gan aon dabht. Ní thuigim cad chuige go bhfuileann tú ag rá ná fuil. 'Sé 'n rud go ndeirinn thuas gur mar /təɪg/ adeirthear é i Múscraí, bíodh go bhfuil /taig/ le fáil in áiteannaibh eile. Cailíocht an chéad ghutha sa bhfocal do bhí i gceist agam.

This is all correct, but you need to add to this that the past habitual/conditional has a -dh that is /x/. Déanfadh, dheineadh, etc.

There are some words where -gh is /g/, eg brugh, pronounced /brog/. A rare word ("mansion").

Caoilte, Dónall Bán Ó Céileachair did an Irish translation of James Murphy's Hugh Roach the Ribbonman. I have both editions. James Murphy used unusually spellings to show Hiberno-English dialogue: “sorra mather” (=it doesn’t matter), “thrubbles” (=troubles), “childher” (=children), “darlint” (=darling) and “crathurs” (=creatures).