If your screen is burnt they don't need to change the motherboard as well, so what they're essentially saying is they'll do what apple does and instead of fixing the phone they'll give you a replacement (at a cost to you).
The choice you need to make is whether to keep your warranty or your photos. If you want the warranty go down that route with LG.
If you want your photos replace the screen yourself or have someone do it for you. I run a phone repair business and can tell you the cost price of the g3 screens hovers between £25-40 depending on how many you buy and where from, so if you go to a shop expect to shell out around the equivalent of £50-60 in your currency. LG will, more than likely, charge you more.
That been said if its more than the screen and if your motherboard has decided to give up the ghost then you'd be cheaper and better off just buying another phone, unless you can somehow use that as a ticket to get LG to replace it for free given most motherboard failures have nothing to do with the user at fault.
If your mobo is dead they won't be able to detect root so don't worry about that.