Your question is one based on a current myth that Sunni's, Shi'ite and Kurds have always been perpetually, three separate "communities", which is absolutely not the case.

Sunni & Shi'ite:

The distinction between Sunni and Shi'ite, arises from the 8th century political civil war, over who should be the Caliph (the Successor), to the Prophet Muhammad. This was not a theological dispute, as the Qu'ran says nothing about this topic, but rather a purely political one. Overtime of course Shi'ite Muslims have acquired certain particularities, but so have different sects of Sunnis, see the difference between Sufi's and Wahhabi's, that is a theological dispute, within Sunni Islam. The difference between the Partisans of Ali (Shi'ite) and Sunni (Orthodox Muslims), really only stems from the very recent past.

Of course there were tensions in the past, specifically in the reign of the Ottoman Sultan Selim the 'Grim', who fought wars against the Shi'ite Safavid Qizilbash, or between the Shi'ite Fatimids of Egypt and Sunni Abbasids (and later Seljuk Turks). The fact remains though, that this was "religious" difference at an elite level, over who could call themselves as the true "successor" to the Prophet Muhammad, vs any notion of theological dispute. Like no one in 15th century Europe would call themselves a Catholic (until the Protestant Reformation, and even then, Catholic/Protestant identity differences remained scholarly endeavors), no one in 1900 Middle East would have called themselves Sunni or Shi'ite.

Kurds:

The third community to dispel this idea of "difference" are the Kurds. The Kurds originate as part of the Indo-Iranian wave of nomadic people that entered into the Middle East in ancient times from Central Asia (see Medians, Achaminids, Aryans in India). The difference being that the "Kurds" settled in the mountains of Eastern Anatolia, Northern Iraq and the Northern Zagros of Iran, maintaing their seclusion from the political machinations of invaders in the floodplains of Iraq. Until the late 19th/early 20th centuries, there was very little to differentiate Kurds from Arabs, as notions of "identity" were not collected based on "ethnicity" but rather on religion, with Sunni's and Shi'ite being listed in tax records as "Muslims". The real separate identity for Kurds comes from their Iranic language, which is different from Arabic (although many Kurds then, and now are bilingual). This difference only began to play a large role, during the short lived Kingdom of Kurdistan, under a local warlord Mahmud Barzanji, who like the Sharif of Mecca, Hussein, was NOT a Kurdish nationalist, but an opportunist, attempting to secure a fiefdom for his own in the hills of Jazaria (the province North of what was "Iraq"), like in previous times. Ottoman reforms in the mid-19th century, had begun to centralize rule in the region (after evicting the Mamlukes of Iraq), at the cost of autonomy of the Kurdish peoples.

The Kingdom of Kurdistan was in response to an Arab Kingdom under Hashemite rule (which never really materialized, except in Transjordan and Iraq), with the many different Kurdish clans coming together in unison, behind Barzanji, in an attempt to secure their own nation-hood, which as we know, they failed to do.

As a fun aside, there was talk of having an American mandate in the rump state of Armenia and Kurdistan, along with the British and French. But the American Congress would have none of it, dropping President Wilson's pretensions of a "League of Nations" (not that Wilson wanted self-determination for non-Europeans to begin with).

Conclusion:

So overall when the British came to Iraq, the idea of uniting these three provinces, was not "abnormal". There was no such thing as "self-determination", until post-WW1. Before the Treaty of Versailles, nations only existed because of battlefield victory, not through treaty and diplomacy (Congress of Vienna was only legitimized because of battlefield victory). There was no such thing as the "Kurdish" or "Arab" people have a "right" to a "nation". And we have to remember that the Arab "independence" leaders (the Hashemites), were not nationalists themselves, but rather opportunists seeking to expand their realm at the expense of the collapsing Ottoman Empire. They had no identity of a greater "Arab" nation (different from greater "Muslim" identity, but that is a separate discussion). That level of dialect did exist to some degree, but would only be strengthened later in the century, specifically in the post-WW2 era.

So when you ask, why did the British do something stupid like put three separate communities into one country, the simple answer is that, this was the precedent since the beginning of time. The British were doing what they needed to do, to secure their access to the Persian Gulf and to the oil supply of Northern Iraq. Why did Prussia do something as stupid as annexing Polish land? Didn't they know the "Polish" were a unique community, who had a "right" to statehood? No. There was no debate about it, they were various peoples, who spoke a different language, who would be ruled as part of the Kingdom of Prussia (and later the German Empire). Same goes for the idea that the British or anybody else had any apprehension about forming a new state from scratch.

Source: Columbia Professor Richard Bulliet, "Modern Middle East" podcast (iTunes U): Middle East in WW1. I can list other sources as well, but Bulliet sums it up the best. Another good book is the "A History of the Modern Middle East", by Cleveland & Bunton.

Under the Ottomans, there were times, including in the 19th century, when the three provinces had previously been administered as single unit from Baghdad, so there was a precedent, although I'm not sure if this was the way they were administered directly before the British occupation of Iraq in WW1.

There are a couple of reasons why the British grouped the three ex-provinces:

1. When they made the Faisal bin Hussein of the Hashemites King of Iraq in 1922, they created a ruling elite which was Sunni and foreign (all of Faisal's men were subordinates from the Hejaz and Syria). They had no local support and they were based in the 'new' capital Baghdad. The inclusion of Mosul brought the Sunni Kurds into Iraq, who made up approximately 20% of Iraq's population and balanced the Sunni population against the Shia.

2. While Basra (which had a major port and agricultural lands) and Mosul (which had oil) could have been economically viable states, Baghdad could not have been economically viable by itself.

3. Financially, Britain was cash-strapped after World War 1. They didn't have the budget to manage three separate provinces-turned-countries.

That's what I remember off the top of my head (wish I could go back to my sources right now). Simply, Baghdad could not be a viable state without both Mosul and Basra.

What you should note however is that the 'stark ethno-sectarian differences' which exist today were not as exaggerated at the time as they are now. King Faisal adopted a Pan-Arab state ideology which permeated the education system and military and which reinforced the Arab identity above the religious. One of the most important Ministers of Education of the 1920s and 30s was Shia, and the Baghdadi Jews were eager, early proponents of Pan-Arabism. (Baghdadi Jews were the largest demographic in the capital, if one counts Sunnis and Shia separately - there were 100,000 Jews and 120,000 Muslims. It was only in the mid-to-late 1930s that Pan-Arabism developed Anti-Jewish overtones, and most Jews left Iraq for Israel in the 1950s.)

Meanwhile, though the Kurds generally did not want to be ruled by Arabs (they were more open to direct British rule), they were an important ally of King Faisal within the Iraqi parliament and voted for his policies. The acquiescence of the Kurds was achieved by 1926, after eight years of intermittent rebellions and the threat of Turkey intervening in Mosul. King Faisal appeased them with promises of respect for their minority rights, and this did work to blunt Kurdish opposition to his government. But the failure of the Iraqi state to follow up on those promises led to strife between the Kurds and the Government again in the ensuing years and decades.

In my opinion, the demographic and geographic makeup of Iraq was not the blueprint of a failed state, and it should not be seen that way, but it did lay the foundations of many of Iraq's political and social issues in the 20th and 21st century.