Desktop vs Laptop
A $1000 desktop for PC Gaming is a good system to start with, and is arguably a better system than a comparable laptop. While the laptop does have massive portability, the desktop has several other advantages that could potentially make it the winner.
portability and size
Starting with the main difference, portability and size, there are lots of different cases that can narrow the odds between the two form factors.
Most Gaming PCs are housed in mid-tower cases, and while these are smaller than their full-tower big brothers, they are still much bigger than laptops.
Micro-ATX mid-towers are smaller than standard ATX mid-towers, and use the micro-ATX motherboard as well. This is a square-shaped motherboard, which offers a different design scheme.
Mini-ITX is smaller still, and can be housed in even more compact cases. Mini-ITX is capable of being housed in a case roughly the size of an Xbox 360. This form factor can still incorporate higher-end components such as closed-loop CPU coolers, m.2 NVME SSDs, high-powered GPUs, and plenty of RAM.
Another factor that can tilt the fight towards the desktop is upgrade/expansion. While a laptop can typically only have its memory and storage abilities upgraded, a desktop can have all of its components upgraded.
A desktop also allows for more things to be added to it internally (though this generally requires a modest case, and a standard ATX motherboard. This area plays directly into system repair costs, as a failing CPU on a desktop is relatively easy to fix, while the same issue on a laptop would result in the purchase of a brand new laptop.
dollar for dollar
While a $1000 Gaming PC (desktop) will be beaten by a high-end PC laptop, that is not a fair comparison. One must keep the costs the same in order to have a fair comparison.
In doing this, we see that the desktop generally is more powerful, as the laptop is forced to downgrade the overall performance, in an effort to minimize heat and prevent thermal throttling.
This is also done to allow for gameplay on battery for a short period. The most important component of a gaming laptop (the GPU), is generally downclocked from the model found in desktops.
A $1000 Gaming PC (desktop) can also be used for other functions easily, such as running as a media server, operating as a game server host, being used as an email server, and a host of other services.
Having the expansion abilities is a massive advantage here, which is amplified by the superior cooling capacity. Heat is the enemy of computers, and must be dealt with swiftly.
While a $1000 Gaming PC might not be the fastest machine on the market, it can easily be upgraded significantly (where a laptop cannot). A desktop also allows for heavy personalization via custom fans, paint schemes, LEDs, and custom power cables. There is no mistaking a custom-built Gaming PC.