Information on a death certificate can be wrong. In Ireland in the 1800s no documentation was required. (None was available anyway in most cases). So the information was only as accurate as the informants knowledge. Ages etc were guessed all the time. You do sometimes see “unknown” written on certificates here but people clearly didn’t like to leave boxes blank and guessed when they probably should not. I expect that might have happened in the US then too. So wrong information on the death certificate is a possibility.
Being married and having children in Scotland would be perfectly feasible. Tens of thousands of folk from Fermanagh and adjacent counties went to Scotland to work, and it would explain why we can’t find any records in Ireland (though not why your Patrick said he was born in Fermanagh on his naturalisation application). I searched for a marriage for Patrick & Anna in Scotland but did not find one. McIntyre is a very common Scottish name and there are dozens of possible children so I cannot say for certain whether any of them may have been born there. I’d need to know where they were living to search for that.
I went through some of the trees on Ancestry again and found several that show some or all the children born in Scotland. At least one has a link to a 1901 Scottish census for Edinburgh. (Attached). The couple in that 1901 census were Patrick McIntyre & Mary Ann Marshall, both born in Edinburgh, and married there 3rd Aug. 1891. So not Anna McManus. All their children up to 1901 were born in Edinburgh. One the face of it, that family’s origins appear to be entirely Scottish.
Rose Ann who went to America appears to have married Joseph James Eagan. Some of the trees have her mother as Anne McManus born Ireland and others as Mary Ann Marshall born Edinburgh. Clearly both can’t be correct. (There are a lot of inaccurate trees on Ancestry, to put it mildly, so this sort of conflicting and unreliable information is fairly common).
Your Patrick was an adult in America when he applied for naturalisation in 1915. He clearly cannot be the son of a couple who only married in 1891. That doesn’t rule out him being born in Edinburgh but I’d be surprised if he was. His naturalization application is quite clear that he was born in Fermanagh, and came to the US from Londonderry in Feb 1900. Presumably the US censuses all have him as born in Ireland too?
You mention a reference on familysearch to Patrick being born in Belfast in 1889. I can’t actually find that reference myself but equally there’s no birth certificate that matches the information in the Irish statutory records. The only male McIntyre born in Belfast that year was a Thomas (who had other parents).
Patrick’s handwriting on the naturalization application is clear and confident. It suggests he was reasonably educated. So more likely to have his facts right you might think. He’s a bit of a mystery though.