This is not a great idea, for the reasons you have noted. Taking the specific question first
is the user obliged to attach the copyright and license notice to the copied files?
It depends on what the licence obliges them to do. Some do require preservation of existing notices; some do not.
As for best practice, the GNU GPL advises people wishing to release software under it to include a single copy of the entire GPL, and at the beginning of each file insert a copyright notice, and text like
This file is part of Foobar.
Foobar is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License, version 3, as published by
the Free Software Foundation ...
You should have received a copy of the GNU General Public License
along with Foobar. If not, see https://www.gnu.org/licenses/.
I have trimmed the text to keep the question manageable, but it only runs to three substantive paragraphs, and is of course not mandatory in form. So best practice is to put a copyright notice and some reference to the choice of licence in each source file.
None of this is an absolute requirement, but presumably you're publishing freely to maximise the use of what you write, and one element of making something maximally usable is the minimisation of doubt about how it can be used ("What's the licence on foobar anyway? Did this file even come from our copy of foobar? Stuff it, let's just write our own