The only potential problem I see is statically-linked applications issue. However, I think in the case where libraries are open and shared amongst many applications/other libraries, it makes no sense to use static linking. Its advantages are that you don't need to worry about dependencies but that "is a solution to a problem that doesn't need to exist in the first place" since package management takes care of these issues.
IMHO it's better to spend 30 minutes once a year resolving an issue that arose when a Debian/Ubuntu repository got screwed up for a short period of time, than to constantly try to solve the problem of managing software installation/upgrades.
However, this comes at a price: You'll always have to provide a newer version of the application whenever a serious issue is fixed in one of your dependency libraries. This means extra work for you. Also, you'll have to take care of an auto-upgrade mechanism.
On systems with a package manager, you'll only have to provide updates if you fix an issue in the application itself. You also don't have to write an auto-upgrade mechanism. Just upload a new package to your server.
However, if you don't care that much about users' security, statically linking is indeed a lot less work than using a package manager.
(FWIW, I'm the maintainer of Mingw-cross-env (http://mingw-cross-env.nongnu.org/). This is a project that provides a cross-build toolchain for statically linked Windows applications. It builds a cross compiler and lots of free software libraries. We have that kind of discussion from time to time on our mailing list.)
But if the applications are all statically-linked, I don't have to worry about dependency conflicts, and some things are a lot simpler, e.g. copying the binaries off to a flash drive to take them to another system, or running two different versions of the same software simultaneously.
On Windows, I don't usually even run installers anymotre; I extract the installers, collect all of the dlls into the application directory, and compress everything with UPX. Once the program is running out of a self-contained directory, I can run it off a thumbdrive, copy it to my Dropbox, etc. Plus I have the option of not updating software when I prefer not to.