Why do launchers not support all widgets? Why is it, that when you attempt to add widgets to two different launchers on the same device, that each launcher seems to present a different list of available widgets? There are always some widgets available to both launchers but others are available only to one or the other.

For example, I have just purchased an LG G3 and quite like the look of its Home launcher. But it doesn't seem to offer shortcuts like Nova launcher does so I thought I could simply recreate the look of the Home launcher in Nova launcher, but most of the widgets that were on the Home launcher desktop by default are not available to Nova launcher.
Some of the below is speculation, or given to me by sources I can't cite so take it as is. 

Some (many) of the OEMs (manufacturers) like LG and Samsung embed their propitiatory widgets within their own launcher. This (allegedly) is down to them not wanting you to emulate their own look and feel in someone else's launcher. 

Samsung, beside Android, have their Tizen Operating System. It is almost identical to their Touchwiz overlay on Android. So if a user has a Samsung, they can use this device without porting over any knowledge, to all intents and purposes they don't know what OS it is running as the interface is consistent. Keeping Touchwiz and those things their own, and unique is a way of retaining their customer base and keeping the UI the same across a range of devices makes it an easier decision to 'get the LG' or whatever to keep what you are used to. The Galaxy Gear 2 Smart Watch actually runs Tizen, not Android as a real world example. 

It's a business decision, not one I am a fan of as I believe in customisation and options. Any UI (ala Touchwiz) that deprives me of choice is one I quickly abandon.   

It is worth pointing out that there are thousands of Widgets on the play store, many of which bear an uncanny similarity to those of the OEMs. 

Last point I'll add, HTC are currently rolling Blinkfeed (their launcher/news stream) onto the Play Store. This may be the beginning of OEMs splitting onto the Play Store as Google did a while back (calendar, keyboard, etc). This may be a way to get some of the widgets you love, without owning the device. However, the locked widgets remain an issue.