United Nations General Assembly Resolution 2758, the legislation that essentially switcheroo'd the national origins of Chinese representatives in the United Nations General Assembly and Security Council mandated three key policies:

• The People's Republic of China, based in Beijing, would be considered "the only legitimate representative of China to the United Nations."
• The Kuomintang exiles in Taiwan would no longer be regarded as representatives of China, they expelled "the representatives of Chiang Kai-shek from the place which they unlawfully occupy at the United Nations."
• Any change in Chinese representation in the UN would need to be determined by a two-thirds vote.

So now that we're clear on what exactly the UN did, I'll address the questions!

Why was Taiwan kicked out of the United Nations when the People's Republic of China joined and took it's place?

It's important to eschew any anachronism of thinking of the Taiwan of 1971 in the terms we'd regard the current Taiwan, a regional trade hub with generally-accepted democratic institutions, a degree of sovereignty and a relatively high human development index. The Republic of China's start on Taiwan was messy, plagued by many of the same crises, power struggles, and human rights abuses that chiseled away at the Kuomintang Party's reputation on the mainland. Even in countries which officially aligned with the RoC ideologically and politically, like the United States, the RoC didn't enjoy much good will.

Today, most consider the PRC and RoC fundamentally different and lacking much cause for armed conflict, you could even suggest they're somewhat coexistent. In the context of the Cold War, however, this wasn't the case. The Taiwan Strait was a naval no-man's-land and both governments stringently claimed legitimate governance of the entirety of China, from Manchuria to Xinjiang to Taiwan. Furthermore, each country's rhetoric suggested that, not only were they hostile to each other, but that they'd regard other countries entertaining diplomatic relations with the other claimant with that same hostility. This position diplomatically isolates RoC as most of the world obviously has a more pressing interest in maintaining relations with the significantly more powerful PRC.

This matter forces the UN's hand, as the body is tasked with moderating diplomatic relations across the globe. Albania got the ball rolling with a resolution and an accompanying memorandum essentially stating:

• Like it or not, the Chinese people, at an overwhelming ratio, are governed by the Beijing government.
• The PRC took Bejing after a Civil War and proclaimed themselves the successor to the old government on the basis of popular support, as opposed to the RoC whose continued existence they considered an impossibility without extensive US military presence in Taiwan.
• In a purely practical sense, the PRC's participation in global diplomacy was more important than the RoC's. They represented a larger population, geographic, and economic interest in global affairs, and would therefore be a more productive presence. They felt, "no important international problems could be solved without the participation of the PRC."

Throughout the process of the debate, the US representatives fielded stringent objections to the Albanian resolution. The US proposed separate resolutions, namely one which would lay the groundwork for a "two China" approach to the issue, and another which would remove the language from the Albanian resolution which necessitated that the Taipei representatives be excluded from the General Assembly. Both of these US resolutions failed, but it is worth noting that the motion which would have allowed both the PRC and RoC representatives to participate only failed by 10 votes.

The Albanian resolution passed at a pretty handy 76-35, with 17 abstentions.

Was there any international push to include both?

Like I said, this was one of the USA's attempted alternative resolutions. One even allowed for a PRC representative to sit on the Security Council, so long as an RoC representative stayed as well. Support for this proposal was negatively affected, however, when the PRC reaffirmed its hardline stance that the Taipei government was no government at all, but an unlawful occupation of a PRC province. The PRC stated it would participate in no international body which also recognized the RoC in the manner which the US proposed. The US attempts at compromise therefore lost a great deal of steam and the more practical Albanian approach gained favor. Alternative resolutions were also sponsored by Saudi Arabia and Tunisia, essentially affirming their support for the US position and denouncing any UN attempt to deny the Taiwanese population's right to self determination. These, however, didn't make a substantial dent in the more generally-held Albanian position that the government which most Chinese people recognized ought to be considered the government of China for diplomatic purposes.

Sources Luckily, the full text of the resolution is available online! [http://www.un.org/ga/search/view_doc.asp?symbol=A/RES/2758(XXVI)]

More specific information on the debates and alternative resolutions is available in the Yearbook of the United Nations, 1971. Refer to section "Questions relating to Asia and the Far East," subsection "Representation of China in the United Nations," starting on page 126. [https://www.unmultimedia.org/searchers/yearbook/page.jsp?bookpage=126&volume=1971]