There isn’t really such a thing as a “generic zigbee bulb” because zigbee allows for different “profiles” so you need to know what profile The device is using.
In the case of lightbulbs, they would normally be using either the ZHA profile or the ZLL profile or the new Zigbee 3.0 profile.
Bulbs which are using the ZHA profile should be able to be used directly with the Smartthings hub, although they may require custom code.
Bulbs which are certified for the ZLL profile are, Per the specs, supposed to fall back to a ZHA profile when joined to a ZHA hub. However, they don’t all do so properly, which is why some bulbs will work with the hue bridge (ZLL) but not when connected directly to smartthings without the use of the bridge.
And some bulbs which are using the new 3.0 profile can only join through “touchlink commissioning” which is a ZLL feature not supported by the smartthings hub.
Based on the comments and the seller responses for the product you linked to, it looks like those particular bulbs will require touchlink commissioning, which is why they would work with the hue bridge but they are not going to work when joined to smartthings directly.
There appear to be some language issues with the seller, and they are trying to tell people that these will need a ZLL bridge, but they are not communicating that clearly, and keep just saying “zigbee” which is not specific enough.
So if you want to try these bulbs, my first suggestion would be to get a Hue bridge and see if they will work with that. If they will, then you can use the official smartthings/hue integration and control the bulbs that way.