I want to stress the most important part of your post if someone wants to buy a pre-owned Mac that it should only be purchased directly from Apple or a very trusted reseller to make sure the Mac doesn't have a firmware password lock and is not still being managed by the previous owner (both of which are very very bad and sometimes hard to detect).
Also make sure you have frequent and regular backups or you will lose your important data -- guaranteed! There are a lot of ways to lose your data besides a hardware failure. Even a simple macOS update (not upgrade) can corrupt the firmware of the T2 security chip (or its equivalent) where to fix the Mac will destroy the data even though the data technically is still in tact, but inaccessible. To add further insult to injury SSDs have a limited amount of data that can be written to them over the entire life of an SSD (typically the max is 150TB to 300TB depending on size of the SSD). If the SSD wears out or fails for any reason the M1 Macs are bricks until you get it repaired since you won't even be able to boot to an external drive (and doubtful if you will even be able to access the data in Target Disk Mode).
My personal opinion of the USB-C Macs is they are not a very good option for anyone, but especially those with limited funds. You will only want to use an official Apple power adapter or you risk causing expensive damage to the laptop using a third party sub-standard charger. You also have to be extremely careful about the quality of the peripherals and accessories you connect to a USB-C Mac as they can also cause expensive damage to a USB-C Mac. Keep in mind any out of warranty repair for a 13" model begins at $500 US and goes up from there. Keep in mind that if something small like the glued in speakers need to be replaced it will cost you $500+ US. With accidental damage you may as well buy a new computer. Consider that a lot of simple scuffs that you may ignore Apple may consider as accidental damage. If you live in a humid region Apple will say your Mac has liquid damage if the liquid contact sensors change colors due to the high humidity.
For people with a limited budget a non-USB-C Mac is a decent option, although hardware support will be mostly non-existent after next year.
Depending on a person's needs a non-Retina Mac is still a good option if you are willing to pay for an SSD and battery (and possibly a memory upgrade). Or go with a non-Apple PC, but stick with a business class laptop for a better quality more powerful laptop that will have a chance of having replacement parts available if needed.
Installing Linux on an older laptop (Mac or general PC) is a great way to have a current up to date free OS with access to lots of free open source software. Once setup and configured the Linux PC will just run for years without issue and it will most likely run faster than it did with Windows or macOS on it. Of course Linux is not ideal for everyone as it does also require a person to learn a new operating system, but Linux is a great way to extend the life of an older computer (Mac or general PC). Linux Mint, Kubuntu, or Ubuntu-MATE are the best options for learning Linux as these all look nice and work on older systems with limited resources and is easy to install (just make sure to have a good backup first).
As for your 2010 laptop the most likely causes of slowness are a bad or worn out hard drive, or some third party software, or not enough memory (most likely to least likely).
Keep in mind my opinion of the USB-C Macs is from personal first hand experiences having to support, troubleshoot, and repair them for our organization. If you knew all the terrible things I've seen with these USB-C Macs you would never ever consider buying one of them. I only listed the easiest to describe highlights of the hidden dangers of these new Macs. It doesn't help that macOS is in a similar decline in some ways. For someone with a very limited budget, I suggest getting a popular model of business class laptop and only buy direct from the manufacturer since what is sold in the stores is usually of lesser quality even though the model may be nearly identical. Inexpensive or consumer grade laptops sold in stores are all mostly junk with sub-standard CPUs, slow memory and drives, and poor build quality. It is so unfortunate that manufacturers clutter up the options with so much junk that makes it hard for the average consumer to find a good quality product.
I'm not saying that everything about the USB-C Macs or M1 Macs are all bad, but there is very little to redeem them in my opinion and experience. It is best to look before you leap.
