I'm lucky enough to have several spare Macs at work. So far, I've upgraded a 2009 MacBook (white plastic), a 2010 13" MacBook Pro, and a 2008 MacBook (aluminum unibody) to macOS 10.15.7 without problems.
In this case, I did it as an experiment and there was no risk as the previous user had upgraded to a newer Mac (i.e., complete drive erase and new macOS install for each of the Macs), but I haven't done any kind of benchmark testing after updating to macOS Catalina. macOS 10.15 seems to have some GUI optimizations that make Catalina feel snappier as well as background security and stability optimizations, so those are part of the reason for me to upgrade.
I would definitely recommend a full backup of your system and apps using Time Machine before installing the dosdude1 patched update. You can also install a full version of macOS on an external drive and migrate your internal drive's data to that system to boot from USB (or Thunderbolt) as a backup system.
The main reason to upgrade macOS is software compatibility: I noticed that the latest version of Adobe apps no longer work on macOS 10.13 this week, for example. Microsoft has officially announced there will be no new updates for Office apps running on 10.13 as well.
Apple itself is even more ruthless: Safari, the newest versions of the iWork apps (Keynote, Pages, and Numbers) as well as free apps like GarageBand, iMovie (and paid apps like Logic, Final Cut Pro, and FileMaker) often require the latest or next-last version (Mojave) of macOS, so it leads to a never-ending hamster wheel of upgrading hardware and macOS. Apple often removes easy access to old versions of macOS and its apps online, but the dosdude1 tool allows you to download macOS from a repository.
The biggest caveats to the dosdude1 patches for unsupported Macs are the AMD Radeon 5XXX and 6XXX series of GPUs. Specifically, these GPU models are in the 2010-2011 iMacs and the 2011 15" MacBook Pros – if you have one of those, you should stay on macOS 10.13 (which works with most contemporary software but support is being withdrawn). They can't be run with graphics acceleration in the 10.14 and 10.15 patches, so they are almost unusable with the new OS versions.
And to be clear, the dosdude1 patches (for 10.13, 10.14, 10.15) do work with graphics acceleration on the rest of the supported Macs, so whether you have an integrated Intel or discrete Nvidia or AMD GPU, it will continue to work with macOS apps with full acceleration.
If you use graphics-heavy applications, support is probably patchier. The major graphics apps and games moved to Metal graphics years ago, so I would hesitate to use OpenGL-era Macs to do any kind of heavy lifting.
The dosdude1 patcher installs updates to address hardware issues after installing macOS: these include fixes for the display backlight, trackpad, USB, wi-fi, iSight/FaceTime camera, APFS booting, Night Shift, and more. Definitely watch the Tutorial Video to get an idea of the install process. Make sure to find out exactly which model of Mac you are working with (for example, a 2010 15" MacBook Pro is a MacBookPro6,2 – you can find the model number for your Mac on https://everymac.com) as the hardware patches are specific to each model.
There are good reasons to keep old Macs running the latest macOS versions – some of them have FireWire ports for example, and not everybody has transitioned to Thunderbolt 3 equipment. Apple removes support for old Macs for specific reasons – old Macs don't have some of Intel's security hardware or SGX media controls (for Ultra HD Blu-ray playback, for example). These checks have been disabled in the patches to keep macOS happy and capable of installing updates, so keep that in mind regarding system security.
If you're thinking about trying the patched macOS update, it might be a good idea to consult a friend who is experienced with Macs. The dosdude1 patches are fairly well known in the community by now, and most people working with Apple computers on a professional basis have probably tried installing them on old Macs. Carefully read all the notes on dosdude1's page (linked below), and also check out the accompanying forum thread at MacRumors (also linked):
http://dosdude1.com/catalina/
https://forums.macrumors.com/threads/macos-10-15-catalina-on-unsupported-macs.2183772/