Does Windows 10 need a page file?

Windows requires that a page file be present, otherwise very nasty things will happen when the system runs low on RAM and there is no page file to back it up.

Should I let Windows manage page file?

No. We highly recommend all users let Microsoft Windows choose the best initial, maximum, and minimum settings for their virtual memory (page file). Disabling or setting the page file size too small can reduce system performance and cause instability and crashes in Windows.