This function provided under the long-press of the tabs button! Alongside 'close tab' and 'new tab'.
I'd also like to point out that the home button is essentially the same as creating a new tab, it just doesn't pull up the keyboard. Common misconception!
Me personally, I'd love to see the home-page's private browsing toggle button moved to where the home button usually is. That'd make the private session just 2 taps away!
As someone who just switched from Chrome I hate the way you have to go through extra steps to get into private. It should be in the menu like you said. Please add this Firefox.