I’ve done some research:
1. Windows 10, Chrome 57.0.2987.110 (64-bit);
2. in my opinion it’s not related to server’s setup: on other browsers – it works;
3. previous instances of WP 4.7.3 – Select Files works;
4. new instances of WP 4.7.3 – Select Files no reaction;
5. it’s partialy Plupload/Moxie issue (https://www.plupload.com/punbb/viewtopic.php?id=14914) – it’s used on that Select Files button by WP core, to generate hidden file input with accept argument which holds accepted mime/file types;
6. workaround #1:
6.1. open (using chrome, win10) wp-admin > media > library > click “add new” (while in grid, not list) – “Drop files anywhere to upload” block appears with Select Files button;
6.2. open chrome developer console, find that hidden file input, it’s right before </body> tag, named with eg. id=”html5_1bblpaq4r1ee0e6ich21tge1mus5″ and it has big accept argument with file types;
6.3. copy “accept” attribute big value to your favourite text editor, find “application/zip”, change it to “application/x-zip”, then copy-replace new big list of mimes/file-types to that “accept” attribute (don’t refersh page);
6.4. click on Select Files – it should work;
7. workaround #2:
7.1. open chrome settings, show advanced settings at the bottom, under Privacy uncheck “Protect you and your device from dangerous websites”;
7.2. refresh media library page and go to Select Files (like in 6.1. explanation) – the button should work.