They used to, but got rid of that policy in 1986 (which led to a split in the ranks and the creation of Republican Sinn Féin by former party leader Ruairí Ó Bradaigh).
That article is a bit misleading, talking about how SF rejects the term “Republic of Ireland” as the “formal” or “correct” name for the southern state. The Republic of Ireland isn’t the formal name for anything except the football team. As the Constitution says: The name of the State is Éire, or in the English language, Ireland.
There’s a distinction, as far as Sinn Fein is concerned. Just as they consider that the British have no legitimate claim to rule in Ireland, so they would argue that the Irish have no legitimate claim to rule in Britain; therefore it would be inappropriate for them to take their seats at Westminster.
The same thinking doesn’t apply in Ireland, where Republican legitimist thinking has focussed on the legitimacy of the particular political institutions in place in the Republic and in Northern Ireland. But there’s nothing fundamentally illegitimate about Irish representatives involving themselves in the government of Ireland.
*Par for the course, but childish and silly. Scots and Welsh Nationalists both take up their seats in the Commons and at least some of them have separatism in mind. As far as I can tell Sinn Fein just want to make a mockery of the system. I don’t know whether any attempt has been made to sign them up to something along the lines of “While it is acknowledged that we seek to remove the Northern Counties from British rule and integrate them into a united Ireland, we shall faithfully discharge our duties to the democratic government in the interim”, or if they’re just flat-out not interested. It’s not as if there wasn’t plenty of Parliamentary business that has sod-all to do with Irish nationalism, and it would be only reasonable for them to attend and put their views since the DUP does. Or, you know, just admit that they’re not interested in British democracy, and not stand for election.
*
They’re not interested in British democracy. There’s no “admitting” about it; it’s pretty much their whole point. Their sole interest is in Irish democracy.
Which is why arguments about attending to deal with “Parliamentary business that has sod-all to do with Irish nationalism” will have zero traction with them. Why would they want to deal with anything that is unconnected to Ireland? Not only have they no interest in this; their view is that as a matter of principle the British should be deciding these things for themselves, and Irish representatives have no right to be involved. If they were ever to change their policy at take part in the Westminster parliament, it would not be with a view to addressing matters that have no Irish aspect; it would more likely be to address only matters concerning Ireland.
And if, as your friend’s thoughts imply, the response is to exclude Sinn Feinif they won’t agree to take their seats, that kind of makes their point for them. If the UK finds that it has to reject or limit the choices of Irish voters to prevent them from returning Sinn Fein members, doesn’t that undermine the legitimacy of its claim to rule in Ireland?
This assumes that Sinn Fein has any interest in stopping Brexit at all. They may see the Border issues and possible meltdown of the Northern Irish economy as positive results that might foster Irish reunification, and if not - sucks to those Ulster bastards, anyway.
And more expensive than just paying the salary and letting them do the constituency casework (which usually takes up more of an MP’s time than debating and voting at Westminster).