If the package purchased is one of the "cheaper" packages, then yes, the mounts as a whole tend to be too light. That is where manufacturers make their money by producing what in reality are inadequate mounts. Not to annoy, but to trim their production costs. The only way around this is to learn to deal with a flimsy mount or change to a more robust one.
The real down side of this is so many people get jack of having their scopes bounce around like a wind vane that they just give up on astro.
Both correct - most budget commercial scopes are almost always on mounts and tripods that are almost useless. Sure they're cheap, but there's a reason.
The worst offenders IMO are the 60mm refractors and 100-115 mm reflectors sold in camera shops and Australia Geographic. These are quite simply a waste of money for several reasons:
- the aperture is too small to show anything that will interest a beginner,
- the mount and tripod are hopelessly wobbly,
- the eyepieces are poorly chosen rubbish,
- they usually have an equatorial mount, which is completely un-necessary for a visual scope.
This is why many who go and try a few scope before buying opt for an 8" dob as a starter scope, or a Schmidt cassegrain package like the Meade Lightswitch or Celestron. These will be useful for many many years and it's fair to say most amateurs have owned one at some stage.
Alternatively when you have a good idea what you want to do, buy the optical tube, mount and eyepieces to match. While this will cost more you will have a vastly superior telescope.
Regarding tripods... You can make one if you are good at woodwork and have the tools.. I made two, both were superb compared to what was commercially available. In my case I started with surplus Karri floorboards, a type of dark red hard eucalypt similar to jarrah. Result was magnificent. One was for an 8" f/7 Newtonian, the other a very tall one for a very beautiful antique 110mm f/15 refractor - an 1880 Thomas Cooke, no less. Modern glues and varnishes will produce a magnificent result if you are patient.