"Debian has three separate places where software is installed"
What user cares where the package installed?  As long as the software
works, what's the problem?

"Our student just wants to run the software."
Good for 'our student'.  However, he doesn't own that machine, now
does he?  He is not responsible for the operation of said machine
either.  He is however, supposed to use the system within the the
policy dictated by the administrator, school, etc.  Being able to run
whatever you want, could just be an issue in some scenarios.  So what
now?  Have the administrator have to black or whitelist an endless
list of packages?  No thx, not for this admin.

"Running anything as root is a security risk. If the Debian package
for AbiWord contains malicious code (or just a simple bug), it will be
running that code as root, with full power to do anything it likes to
the machine."
Well, if you can't trust the software you are installing, much less
the system it is being installed upon, well, I'll just leave it at
that.  (so really, a non-issue)

"APT relies on a database to keep track of what's installed and what
isn't. This database must be kept in sync with the filesystem... if
the user deletes a file to save space, then APT will continue to think
that the file is installed."
Deleting a system installed file would require the user to have root
access.  If this is the case, there is no system level security at
all.  (yet another non-issue)

"APT often downloads more than you need. Some packages have been
split, for example 'python' and 'python-doc', but most packages
require you to download a considerable amount of data that you simply
don't need."
Eh?  Methinks he's confusing debian/dpkg systems with some fairly
braindead, albeit popular, bloated packaging system.  (swing and a
miss)

"Despite trying to download every file for every feature of a program
you might possibly need, APT still often fails to get things you want.
For example: install gqview and open an image. Choose 'Edit in Gimp'
from the menu, and you'll get an error complaining that Gimp isn't
installed."
That's why there are things like apt policies to install suggested
packages, or even auto-apt.  (omg u loose agane!!!!11)

"APT is not scalable:  Since every package is installed as root, every
package must be carefully checked by a trusted Debian developer."
Really now?  News to me.  Last I checked, debian, maybe ubuntu had the
most packages available of any linux distro.  And again, security
actually matters to some people.  Trust the debian team or trust ...
whoever.  Riiiiiiight.  (sorry, no dice)

and then they go on to say ...

"Anyone can make software available via Zero Install. Trust is for
individual users to decide, not the admin, since their choices only
affect them."
Oh, if that were only how computer security *actually* worked.  Yes,
sometimes user code can be used to exploit root level vulnerabilities,
even under linux.  Nice try, though.

and then, here: http://zero-install.sourceforge.net/filesystem.html
they describe how applications are cached, so that multiple users on
the same system only work off of one instance of a program in the
zero-install cache.  So, exactly how is that only affecting one user
again?  Right, it isn't.  (BZZZZT!)

"APT must download the latest package listing for the whole archive
before doing anything."
Well, I have news for you.  If you're on a slow enough connection that
you can't stand waiting for the packages list to download, just wait
until you try to install any software, either via apt or zero-install.
 Yeah, go "zero-install" open office. (no cookie)

"Upgrading is very slow ... it requires downloading a vast amount of
software, most of which won't be used before it's upgraded again."
Oh yes, much better to run a system where the older software packages,
the ones with possible security issues, stay on the system the
longest.  Brilliant!

Ok, I'll just stop there.  Yeah, it's a really slow day at work ;-)