"For instance, consider the grand mufti of Jerusalem, Amin al-Husseini. He was the leader of the Palestinians in the ’30s and ’40s, prior to most of the history we’re talking about that has so enraged the Palestinians. Nevertheless, the man visited Auschwitz in the company of Himmler and aspired to have his own death camps created in Palestine to exterminate the Jews. He was a full-blown Nazi collaborator, and the head of the Palestinians. As late as 2002, eight years after winning the Nobel Peace Prize, Arafat praised al-Husseini as “a hero.” This is the kind of thing Israel has had to deal with continuously."
Al-Husseini's role in the Third Reich has been the subject of a good deal of politically-influenced embellishment where it becomes hard to disentangle myth from fact- /u/commiespaceinvader's has done an excellent job summarizing the Mufti's storied career in this answer.
For the most part, the historical record does not bear out Harris's picture of Palestinian history. Firstly, although al-Husseini was an important political figure in the interwar period, he was far from the only one and to call him the "leader of the Palestinians" is to impose a political unity that was not present in the Mandate. There were other Palestinian national leaders, but al-Husseini's activities in the Third Reich has led to him being elevated above others, some for legitimate scholarly reasons, others to delegitimize the Palestinian national movement (tellingly, the essay on Palestine in the anthology Arab Responses to Fascism and Nazism Attraction and Repulsion is entitled "More than the Mufti: Other Arab-Palestinian Voices on Nazi Germany, 1933–1945, and Their Postwar Narrations" underscores the outsized importance of al-Husseini and how some historians are tired of it).
The charge that al-Husseini visited Auschwitz largely comes from one source, Dieter Wisliceny, and the linked answer from out almighty moderator does show why Wisleceny is rather problematic source. Harris's invocation of Auschwitz is indicative of a wider problem when it comes to popular accounts (i.e. journalists, intellectuals, etc.) that examine al-Husseini. The Holocaust was an incredibly complicated process both in its genesis and execution. There is a marked tendency to assign blame in some way to al-Husseini for the Holocaust, whether through refusing to allow Jews access to Palestine or giving Hitler the idea for genocide. Harris is dancing very closely to this line of argumentation, but the reality was al-Husseini was an incredibly marginal figure inside the Third Reich. Neither Himmler nor Hitler needed much prodding for genocide, especially from a non-European.
Finally, by not bringing out this wider context, Harris is presenting an incredibly skewed picture of al-Husseini's activities within occupied Europe. After the failure of the Iraq revolt in 1941, al-Husseini ended in Europe and worked for the German propaganda machine. But in doing so, he had to compete with other anti-British Arab emigres and navigate the various factions within the German state apparatus. Al-Husseini's anti-Jewish sentiments put him in ideological resonance with the SS, but in light of other emigre competitors like Rashid Ali al-Gaylani, making an alliance with this particular power bloc had a certain political logic behind it. Al-Gaylani's more secular and pan-Arabist politics had found support both in the Italian and German Foreign Ministries and he was in many ways a more "current" leader against the British, having participated in the Iraqi revolt in 1941. Al-Husseini's main stand against the British had occurred a generation earlier in 1929. Making the most of his connections to Himmler was a way for al-Husseini to gain the paltry resources the Axis devoted to the Arab cause away from an emerging rival.
The above does not to excuse nor does it mitigate al-Husseini's wartime activities. Now al-Husseini very much fits into the mold of being a bad man and a fool, but looking beyond his propaganda broadcasts there was a discernible desperation in his activities for the Third Reich to recognize the cause of Arab independence and place him at the head of it. This was a step that Hitler was unwilling to make. He helped in the recruitment of Bosnian Muslims in the SS Handschar Division, but he was often quite frustrated that the Germans did not put much weight behind his efforts and the Handschar was not used to fight for Arab independence.
So while al-Husseini's activities in 1929 and the mid-30s put him at the starting point for Palestinian national activism, when you remove the moral revulsion from his wartime activities in Germany, al-Husseini emerges as a rather pathetic figure. As an exile, he found himself having to gain the favor of two states whose leaders who at best only supported his cause for tactical reasons. Neither Mussolini nor Hitler saw much use for the Arab world beyond the immediate tactical use of Arab nationalism to disrupt the British. By hitching his star to what was soon apparent to be a losing power in occupied Europe, al-Husseini was out of circulation with the wider developments within the Arab world. Al-Husseini had to reconstruct his network of political contacts after the war as well as deflect charges he was a German collaborator. So in short, al-Husseini's dalliances with National Socialism was more damaging to his political career and although he may have aspired to be "a full-blown collaborator", his would be allies in Italy and Germany were mostly indifferent to him.
The history if generally sound, if framed peculiarly. Starting by discussing Palestine after "the '30s and '40s", as Harris does here, ignores quite a bit of history that is at the core of the Palestinian sympathies with Hitler, Mussolini, and the Axis.
During World War I, the Arabs supported the British and French against the Ottoman Empire, of which they were subjects, with the understanding that they would attain independence in the aftermath. The British and French did not deliver on this, and in November of 1917 the Balfour Declaration was published, which stated:
His Majesty's government view with favour the establishment in Palestine of a national home for the Jewish people, and will use their best endeavours to facilitate the achievement of this object, it being clearly understood that nothing shall be done which may prejudice the civil and religious rights of existing non-Jewish communities in Palestine, or the rights and political status enjoyed by Jews in any other country.
The Palestinians were, obviously, not interested in having a "national home" for anyone set up within their lands, and objected to the Balfour Declaration, as did the rest of the Arabs. This is above and beyond the objections to the partitioning of Arabia and subjecting them to the British and French mandates.
It is true that Husseini met with and proposed alliances with the Axis, first with Mussolini then Hitler. He did this in hopes of establishing independence for his people, as a victory for the Axis would end the British mandate. I have read no indication that he sought to have death camps of his own in Palestine. He was, however, very outspoken about not allowing continued Jewish migration to Palestine, which has caused some to blame him for the Holocaust. The argument goes that "Hitler just wanted to be rid of the Jews. If it weren't for al-Husseini's refusal to allow them to be shipped to Palestine, Hitler would have never opened the concentration camps in the first place." This is absurd on its face, if for no other reason than because al-Husseini did not arrive in Europe until October of 1941, while the Holocaust began in the spring of that year, and internment of the Jews and other "undesirables" in Germany had been going on for two years prior to that.
It is also true that Yasir Arafat did praise al-Husseini, because Husseini was a leader in the early Palestinian nationalist movement, just as Arafat was the leader of the current Palestinian nationalist movement. What Harris is attempting to do here is to use guilt by association, pointing out that Muhammad Amin al-Husseini allied with Hitler, and thus he must be an irredeemably bad guy. And since Arafat praised Husseini, he must also be bad, thus the entire Palestinian nationalist movement is bad. The reality is that this is a very complex situation, and Palestinians at the time felt badly abused by the British and were happy to ally with their enemies. You may make your own judgement call on that front, but it is important to recognise the full context of how those decisions came to be.
For an even longer reading on the subject, I highly recommend Eugene Rogan's The Arabs: A History and John Judis's Genesis: Truman, American Jews, and the Origins of the Arab/Israeli Conflict.