However, usually lasers come as a point, which means that we'd have to make the laser head move in quite an exotic curve, which will slow down our cut rate.
The limiting factor is usually the cutting speed of the laser. This depends on the laser power, the material characteristics (if it even can be cut by laser) and the material thickness. The fact that you have an exotic curve doesn't really enter into it much unless the motion can't keep up. In fact an exotic curve should be quicker to cut than a load of straight lines with sharp corners as the motion needs to decelerate to a stop in each corner.
Is there a known solution to this? In my head, if the beam was instead focused as a blade and not as a point this would solve this, but I'm unable to find any existing solutions.
If it's "focused as a blade" then the energy density will drop and cutting time will increase.
Ideally we don't want to move the laser head, or move it as little as possible to get our number of cuts per second as high as possible.

Figure 1. Flying optics. Image source: Smoke and Mirrors.
You have two options that I know of:
- Flying optics. The laser stays put and the mirrors move around on an X-Y motion.
- Galvanometer mirror deflection but this will result in angled cuts if your material is thick.