In the default version of Chrome 77, it is already a default available function through which users can move NTP shortcuts and set their specific order according to their preferences. It is reported that this change also applies to Mac, Linux and Chrome OS.
If you don't like the animation or quick link grid layout provided on the new tab, you can access chrome://flags/#grid-layout-for-ntp-shortcuts and turn it off temporarily.
Although the logo can still be found on the Chrome://flags page, considering that it has been released as a feature, it may disappear in future Chrome versions at any time.
For reference only: Chrome 77 was released on September 10, 2019, and the next version will be released on October 22, 2019.