Yes, it is possible to form an L shaped desk without the supporting leg in the middle. As you already figured out, you’ll need to find ways to reinforce that middle section. Mending plates are one way but you’ll need a whole bunch of them. And the LINNMON is not solid all through, so you will need to find the right spots for your screws to bite.
IMHO, the wider the surface area the plates cover, the more stable it would be. An alternative is to use the ALGOT metal shelving unit, like how it’s done here in this T-shaped Partner Desks. The ALGOT shelf holes line up rather well with the original holes in the LINNMON, which is a bonus. Check out the video of how Bryan made it.
The truth is, the corner will never be as stable as having a leg there, so be careful not to put too much weight on it.