Since it hasn't been mentioned yet... Yes a failing drive will significantly slow down a computer. Drives are built to be fault-tolerant, so if it reads a block of data and that doesn't match the block's checksum, the drive will attempt to re-read the same data until it gets what it believes is correct data, or until it gives up and sends a failure to the computer.

So now imagine your drive is in a state where nearly every block is having trouble being read, so it re-reads each block several times, adding a significant amount of time to every operation. A scan of the drive may indicate everything is working correctly if the drive does eventually return valid information, but the drive itself is having to work very hard to get this data.

One thing you might try to check for internal errors is running a read/write test of the drive, and recording the speed these operations were performed at. If that number is close to the parameters of the drive (you can check with the manufacturer or online reviews to find real-world drive speeds) then the drive is probably ok. However if the test is running a lot slower than the expected speeds, it's a good bet that your drive is failing and you will want to back up the data as soon as possible.

Is the hard drive failing and constantly blocking reads and writes? Maybe, check the SMART data.

Is the cooler full of hair/fur/dust and clocking the CPU down? Maybe, look at the thermals etc.

If you don't diagnose issues you are just guessing and potentially wasting money replacing parts that are fine.

I hate to be "anti-environment and pro-capitalism", but... just buy a new one. The worst thing about HDDs (in my opinion) is that it's hard to tell when they'll fail. You don't have this problem. Clone the drive, spend $20-$50 on the same storage size, and you're done!

Or double the budget, and upgrade to an SSD. I know it's easy to suggest someone else spend their money, but that's really the only option here...

Even if your hdd is ok, ssd is much faster, first time you use your laptop with os installed on sdd, it'll be new and pleasant experience for you that's for sure, buy cheap one for the first time, 240-256gb is pretty cheap nowadays, and enough for casual web browsing cinema watching and office tasks

100% yes if it's failing. Buy an ssd and revel in the new speed, don't do another spinning disk.

You can check the data the drive itself keeps about disk errors. This data is called SMART.

https://www.howtogeek.com/134735/check-ssd-or-hdd-health-with-smart/

Figure out if the drive is having problems or not, then look at replacing it.

Another thing that slows down old laptops is thermal throttling. Once again there are plenty of programs you can download to check component temperatures. Have you cleaned dust out of the inside of the laptop and re-done the thermal paste?

What size do you need?

What's the RPM of the drive, other specs of the computer?

I’m a bit late to the party but if you’re still running a machine with BIOS rather than uefi You should definitely look into spinrite 6.1 from grc.net written by Steve Gibson who is nothing short of a genius when it comes to low level hdd optimisation. (7 will support uefi) if your machine has legacy support you might be able to get it booted as well. He has all sorts of utilities including ones to make sure you can spinrite will boot on your machine before purchase

(Shields up ring any bells to the oldies, same guy)