West Jerusalem has been part of Israel since it was founded in 1948. East Jerusalem was part of Jordan (then Transjordan) until 1967 when it was captured by Israel in the 6 Day War.

Palestinians claim Jerusalem as their capital. Part of the problem for Palestinians is their refusal to accept historical ties to the land by Jews - and their persistent revisionist history in denying Jewish ties to historical sites in the territories - including Jerusalem and the Temple Mount.

As for the poll, while a majority or near majority want to see a Temple rebuilt (it’s aspirational in tone - considering that the location is the most sacred spot on the planet to the Jewish faith), the poll was far more cautious in whether they would ever see it happen:

The public is about evenly split on whether they believe it will happen, with a slight edge – 42% to 39% – to those who believe the Third Holy Temple will be rebuilt.

Should the State of Israel take active steps towards the reconstruction? Forty-eight percent said no, while 27% said yes.

Not that the aspirations wont be seen as a direct threat to what Muslims consider their third holiest spot (a problem that happens all over the world as one religious group captures a site holy to another).