Sharepoint - Display New Form of a List in a Web Part page

So it turns out that SharePoint Designer is the only way to accomplish this. You have to go into the page in Designer, edit the page, and select the Insert tab from the ribbon -> New Item Form -> select "CUSTOM LIST FORM..." (not one of the pre-populated lists or you will get the barebones default content type!) -> Choose the list for the form you want to show and the content type, click OK -> Save the page in SP Designer and it will now show on the page embedded as a form. Success!

What if you just add ?IsDlg=1 to your link in the page viewer, then it will remove most of the "UI".

/servername/site/Lists/listname/NewForm.aspx?IsDlg=1