As an old-timer who started using Macs in 1985, I used to lament the "loss" of the trash can on the desktop, but I came to realize that 1) it does not make much sense, and 2) there are far easier ways to send things to the trash (e.g. Command-delete, or control-click and choose Move to Trash) that don't require seeing the Desktop or dragging all around the screen.
But I am not one to turn my back on a challenge, so here is a solution for you.
All you need is a simple Automator application, and dialabrain's suggestion to paste an icon onto it (this time it will work, because we're not going to to put an icon on an alias but on the actual application).
Here is a screenshot of the Automator application. It is extremely short, just one action. You want to it to appear on the desktop, so save it to, er... the Desktop.

As dialabrain suggested, copy and paste the actual Trash icon onto this, using the Get Info window.
You will be able to have a desktop like this (note: in my case, at first the app still had the automator icon; the trash icon only appeared after I logged out and back in)

Limitations of this method: the icon is static, it does not reflect the status of the trash being empty or not; and you can't control-click to empty the trash.