The Lenovo Thinkpad X13 G2 was the direct competitor to the HP 830 G8. Fully working used ones sell for about the same price as each other on ebay - in the region of £160 to £200ish.
I personally prefer the HP 830 G8's due to them having slotted in RAM instead of soldered RAM.
Thinkpads used to have really nice keyboards compared to other laptops. But with the X13 G2 they used a short travel keyboard, which I'd rate as worse than the keyboard on the HP 830 G8.
If you want something more portable, go for a 13" laptop. If you want a good compromise between portability and screen size, go for a 14" laptop. If you'll be carrying your laptop in the boot of your car with short walks to your home or "office", or if you have the build of a heavyweight boxer, get a 15.6", 16" or 17" laptop.
HP 840 G8's or 845 G8's are sensible 14" laptop buys for under £200 for a fully working one.
A fully working 16 GB RAM Lenovo T490 or T490S would be a sensible buy at £120ish. They have good keyboards.
For your old laptop, make sure it has an SSD instead of a mechanical hard drive - which from the sounds of things it does, as you've not complained about slow speed.
For the thermals, it might be worth trying to "repaste" your CPU. Over time the thermal paste between the CPU and cooling fins can need replacing. Genuine Honeywell PTM7950 thermal pads are great for lasting longer than traditional paste, and for the cooling to get slightly better with age.
Also a few tweaks to the bios and operating system setting can help a laptop to run cooler. You're basically telling the CPU to run at a less powerful level, which is usually fine, as most of the time for most users, the CPU isn't the bottleneck. So much so that you'd probably notice no difference in laptop speed with the lower powered settings.
And blowing or sucking the dust out of the fan area might help, if it's really clogged up.
I think it would be really nice if you could spend nothing, or next to nothing (about £10 on the genuine thermal pad) to get your 7 year old laptop good enough to give you another year or two or three of faithful service.
BTW, I know someone that's starting their 2nd year of uni. She has a Dell 7470 that was bought used for £170 4 years ago. It's a 7 or 8 or 9 year old laptop now. She finds it to be a great tool for everything she needs to do on it.
She was given an iPad as a present. She takes the iPad to lectures to doodle on the handouts as the lectures are going on. And she uses the laptop for typing assignments, emails, web-browsing, video meetings etc.