I have a new installation of LibreOffice 5.2 on Windows 10.
If I start an Opendocument text file from the main LibreOffice desktop icon I have no font previews in the font dropdowns. I have font previews turned on.
If I start a document from a newly created file on the desktop I have font previews in the dropdowns. If I open a previously created document I usually have font preview.
About the only time font previews do not show is when starting from the desktop Icon and choosing a text document.
I assume you have gone Tools > Options > OO > View …, and ticked Show Preview and Show History.
What fonts do you see when you go to C:\Windows\Fonts?
r-click the desktop icon > Properties. What exe file is being started by the icon?
There was aproblem with font preview depending how you start LO as you diecribed it. In version LO5.3 it is fixed. Try it out.