Ignoring the package will also remove the syntax highlighting. There are two options to do what you want. The first option is to find all the *.sublime-package files, extract them (they are just renamed zips), remove the snippet files, then save the packages as something other than there default names. Then, add the default packages to the ignored list. You can either package these up as *.sublime-package files and add them to the "Installed Packages" directory or move them as a directory into the packages folder. The second option is to override the snippet files. If you take this route, I recommend using PackageResourceViewer. After you install the plugin, open all the snippet files you would like to modify. Then remove the tab trigger and save the file. This will place the file in the proper location to override the built in snippets. You may need to restart ST after doing this for the changes to take effect.
Both are less than ideal, but will give the behavior you want.