Trailer with fixed coupler so it doesn't pivot

Please be gentle with me as I'm fairly inexperienced with towing. I have a small utility trailer with a size small enough that I find it almost impossible to see in my mirrors when backing. Unable to see how it's pivoting, I'm nervous about attempting to back unless I have a spotter (which I don't always have).

So what I thought might exist out there is some sort of trailer coupler that works like a regular pivoting ball mount most of the time, but that can be "locked" in to place when required, sort of making it like a non-pivoting extension of the vehicle it's attached to. That way I'd have less to worry about how it's pivoting while backing because I'd know that wherever the corner of my vehicle is (which I can see in the mirror), the trailer is right behind it on the same path.

I've been looking in various forums for days and everybody gives me bewildered, "uh, why would you want to do that?" replies. Maybe what I'm thinking of doesn't exist? If that's the case, maybe somebody can educate me on why it would be a bad idea? About the closest I can find so far is something like an anti-sway coupler where you could tighten down the sway bars hard enough that you could lock out most of the pivot.

If anyone has advice, I'd appreciate it.