Is it undemocratic that a person with fewer votes than another can become Taoiseach

Let's not pretend it's a perfect system either:

This is true but in many ways it's not ideal either. The fact that very few TDs are assured of getting elected in the next general election means that they are all hyper aware of local issues. This has 2 downsides:

1. They often behave more like local councilors then national legislators. Some of the independent TDs, in particular, don't even pretend to care about national issues and don't bother attending committee meetings or even asking questions in the Dail. Instead they'll devote their issue to dealing with things like fixing streetlights and making phone calls to the passport office for their constituents.
2. A local concentration of voters can have an outsize affect for an issue that only affects them - even if it's for the greater good of the wider area. A classic example is a residents committee opposing new housing in their area. Any politician that they contact generally will be supportive out of fear of losing a potential cache of votes at the next election. Every single party is guilty of this.

There's an argument that this is preferable to a system like ours where you have a collection of people who, more often than not, don't have any specialist knowledge, end up overseeing the running of a government department. A classic example was the appointment of Simon Harris, a man who dropped out of college in first year, as the minister for Higher Education.

Now, obviously the US system is open to patronage (e.g. Trump appointing Mitch McConnell's wife to be Secretary of Transportation) whereby you also end up with unqualified people running departments. However it's easier to find suitable candidates if you're not restricted to a small group of people, many of who's main talent is being electable.

People asking questions like the OP I think are generally coming from a place where they are too embedded in UK or American political media and for whatever reason seem to think their electoral systems somehow are more representative or "fairer" showing a complete ignorance of PR-STV. Like another person mentioned a lot of the recent ignorance to PR-STV is thanks to SFs utter mismanagement of their ballots which if they had done correcrtly they honestly might have come away with a majority of seats by themselves. However they didnt and simply got the most seats leading to this silly idea that they "won" the election whereas the parties who "won" the election are actually those who can cobble together enough seats to form a Government.

My favorite statistic to show those who don't understand this is a comparison for how unrepresentative FPtP is that the UK has in the last 100 years only have had 1 Government elected by the majority of the people ie popular vote and ironically it was a coalition which is more often than not what happens due to PR-STV. PR-STV is without a doubt the most representative and democratic voting mechanism available, is it perfect? No but its far far better than any of the other options.

We don't vote to elect the Taioseach we vote to elect TDs.

This isn't Murica.

This is a very sad reflection on the lack of knowledge of how our Democracy works.

Easiest way I can think about it. 3 seat constituency, 4 parties.

Party A gets 1.5 quotas.

Party B gets 0.8.

Parties C & D 0.6

If Party A runs two candidates. Scenario 1 they split the vote badly. First candidate gets 1.2 quotas second gets 0.3. They only get 1 seat. But they top the poll.

Scenario 2 they split the vote well, both candidates get 0.75 quotas. They get 2 seats but Party B top the poll.

So that's why I don't think topping the poll is that important.

This is the beauty of the PR system. It ranks the candidates in order of popularity. And being "popular" is absolutely not about how many first votes you get - it's about how you are ranked by everyone.

Take a simple system where you have 3 candidates, A, B & C.

A & B are both equally loved and loathed by exactly half of the population.

Candidate C is liked by everyone - he's not as loved as either candidate, but also not loathed either.

On ranking across the population, who is the "most popular". Well, clearly it's candidate C. More people like him than either A or B.

We tend to think that just because lots of people really, really like a candidate than that somehow makes them more popular than someone else who is just "mostly liked" by even more people. That's how celebrity works, but it's not how a democracy should work.

The Taoiseach, Tainaiste and Minister for Finance must be members of the Dáil. Article 28 section 7.

Only two Senators can be Ministers at a time too.

Bless our constitution authors for providing us with the PRSTV. It gives a most fair representative voice to the voters.

Politicians don't like it and twice had referenda (rejected) to replace it with the FPTP in 1958 and 1968

This is such a niche gripe - I love it!

While we're on the topic it always bothers me the way that they round up or down when calculating the transfers of the surplus. Even in that pdf they use an example where all the numbers divide neatly but in reality that's often not going to be the case. In Stormont elections they factor this in and use decimal places where necessary.