Had to look this one up. My knowledge of Irish government is limited. Explain why. https://en.m.wikipedia.org/wiki/Taoiseach
Is the key vote?
It’s not the “key vote” in the sense of tiebreaker or decider etc.
The EU is a confederation of independent countries that pool together some things like trade policy (which gives small countries like Ireland, Estonia etc. a bigger voice and negotiating position in trade).
Now the key point of any confederation is how well it takes care of its smaller members.
the EU, outside of the UK, France, Germany and Italy is all small countries and they were watching Brexit very, very closely.
Brexit mostly affects Ireland - and Ireland would be hit by 1. trade problems no matter what kind of deal or no deal (20% of their business is with the UK, down from 70% 40 years ago, but still a lot
2. by food/electricity (the British minister threatened to cut off food supplies. But this is now solved by increased trade going from the european mainland to ireland directly and also electricity cables connecting Ireland to the EU-wide grid)
and
3. by violence IF a border came up.
Germany and France wanted a deal, the UK wanted a deal etc. - but the former could not throw Ireland under the bus.
The ONLY stickling point in the May deal was around the backstop, which was put up to solve the problem of “no border in Ireland”
So Ireland has been setting EU policy towards the Brexit deal. Think of it - otherwise, there is no issue for the UK