Laptop advice - repair/scrap/cry?

If you aren't particularly technical - and the fact that you took the item to a repair shop, rather than investigate and repair it yourself - then I don't think it makes sense for you to try to attempt the repair by "following YouTube".

If your machine refuses to turn on, it could be motherboard failure, but it could be a few other things - e.g. faulty power adapter, something not seated correctly inside the laptop. It might be worth a visit to another repair shop to see if they can problem solve it more successfully than the original one.

Ultimately though time has a cost, and it's a 4-5 year old laptop, which is a reasonably average lifespan for these things.

The first place is offering you data retrieval for £45 which is fair enough, you could probably do this yourself by buying an enclosure to stick the SSD from your old laptop into, but it wouldn't be cheaper by much when you factor in your time.

I think it's worth visiting another repair shop to see if they concur, but if they do, then £80 for a dud laptop isn't a bad offer, I'd take the cash and put it towards a new machine.

It's great to see everyone is a repair expert on here and suggesting it'll be an easy fix costing a "few pennies".
£80 scrap is disappointing, but it's a reality you'll have to face. Unless the data is on chips soldered to the motherboard I doubt you'll need to spend the £45 on data recovery. If it's a regular SATA interface you can simply remove the drive and fit it in an enclosure to have access to your data.
Most of those repair videos you see on YouTube are just blind luck, very few have any knowledge or understand how to reverse engineer the device to find the more elusive faults.
There's also the cost of the tools to perform troubleshooting and repair - if you don't already own them it's a bit of a cost, never mind the learning curve to use them.

Nice business of laptop and computers

Buy new laptop from this site, transfer data from old laptop to the new one and sell the leftovers - sounds like the most economic approach

Make sure the power supply is good (multimeter) and if so remove the battery and try to power on with power connected.

You could easily get an i5 laptop with 8gb of ram on ebay for about half what he wants so I'd avoid his kind offer.

A 4+ year old, £500 at new laptop, I'd have already stripped it for parts and binned it at this point.

In my exoerience processors almost never fail. Especially modern ones. There are lots of protections built into the motherboard, and cpu itself to stop it getting damaged (over voltage, over heating, fan failure etc). It's more likely something else is the problem. It's possible the mb could be faulty. But again it's more likely to be something else.

Only you can decide if the machine is still fast enough for your needs, and if you want to pay for a repair or upgrade. As for data recovery, Unless your hdd/ssd failed before it stopped working. You won't need data recovery. Just access to the drive to copy data to another.

If you have the the knowledge, experience and tools to repair it yourself, why take it to a shop? I guess you don't, so don't risk it. You'll probably do more harm than good.

It's possible the shop is trying to rip you off (there are some cowboys out there), but I have no idea for sure. If your unsure, get it back and take it somewhere else. Even if you don't get it repaired get it back, it's yours, unless your willing to sell it for scrap. Check ebay to see if any are for sale and what they are listed for or what they have previously sold for to get a rough 2nd hand value. Or check what cex or others would give you.