For 550€ I personally wouldn't buy a new laptop, I'd either buy a used/refurbished laptop or build a desktop PC, cheap new laptops tend to be not so great.
If your existing laptop is still usable, you'd at least have a laptop even if you went for the desktop PC.
For normal use the experience isn't much different, but a desktop is nicer for tinkering and somewhat better upgradeable (RAM, large disks, adding a GPU if it becomes necessary), and if you do calculation-intensive things most laptops run into their thermal limits pretty quickly, a desktop can maintain its top speed.
Sidenote: If your laptop is overheating it might be related to dust accumulating over the years inside the laptop in the area of the CPU, CPU fan, air inlets and outlets. You could probably get it cleaned in a local laptop repair shop or do it yourself (with the risk of damaging something).
If your existing laptop is still usable, you'd at least have a laptop even if you went for the desktop PC.
For normal use the experience isn't much different, but a desktop is nicer for tinkering and somewhat better upgradeable (RAM, large disks, adding a GPU if it becomes necessary), and if you do calculation-intensive things most laptops run into their thermal limits pretty quickly, a desktop can maintain its top speed.