Should I accept this help in fixing a 1" hole in my radiator?

Do it right the first time.

In the best case the "quick and dirty"-fix fails at the first pressurization, in worst case when you need it at most.

What should concern you more: How does a hole in the radiator appear out of nowhere?

Aluminum foil is at best a very temporary fix to get you to the mechanic, it's not a long-term solution. Your mechanic is right, the hole shows that there's damage to your radiator, probably due to corrosion, meaning you will get another hole probably sooner rather than later. Radiators corrode from the inside out, so you may have nice chunky flakes in your cooling system waiting to clog up important channels and lead to much more expensive problems.

You will have to spend the money, better to spend less now and have a reliable car.

This may not be a popular answer, but it's not an immediately critical component, and there's a good chance that a kludge will fix it for years. It wouldn't be particularly unsafe if it failed again, though having it fail on a highway may be jarring. You'll just have to get in the habit of watching for puddles in the unlikely case it leaks. Yes, if you want to do it right, your best bet is to get a new radiator, and flush the old system first. I don't know if I'd pay a mechanic to do much else in your situation.

It would also help to know what year the vehicle is, and the make/model to see if others have experienced this failure and what may have caused it. If this is a 20+ year old car its possible the radiator simply failed from mechanical stress of heating/cooling, and/or corrosion. If it's a newer model it could be a known defect.

JB Weld "extreme heat" and some metal flashing from the hardware store would be superior to aluminum foil which may eventually blow out from the pressure unless you really gooped the heck out of everything on and around the hole. Scrub around that hole with a metal brush and acetone first and follow all safety procedures.

Or, if your radiator is brass and not aluminum or plastic, then you could do like I did years ago and just get a disposable propane torch and a metal coat hanger and just braze the hole shut. That radiator lasted for many years and never had a problem with it thereafter.

The success and longevity of such a repair depends on the material where the hole is. It may be aluminum (rare as a top of a radiator), brass or copper (in older/retro cars), polyolefine plastic (the general case for modern cars), stainless steel (some aftermarket radiators). One must use the proper glue for the material in question.

Elevated pressure (e.g. from a blown head gasket) and/or elevated working temperature (failed fan, thermostat, clogged cooling system) of the coolant should be dealt with as well.

If you know what you do, releasing the coolant pressure by removing the pressure cap may or may not help.

The general success rate of patching a radiator is low. Metal radiators can be reliably welded or soldered (provided proper experience and equipment), but your one probably has plastic top.

The success stories generally range from "I patched the hole somehow, got home and then to the mechanic without issues" to "I patched the hole somehow, sold/disposed off the car soon after".

Many's the time I've fixed radiators like this - with two pack filler, as a get me home, that lasted several years! That's in UK where it rarely gets hot hot. Sealing a stone hole or small seam crack it works well. Of course there's always the danger it may blow again - a weak seam may be what excess pressure from a bigger problem was looking for for an escape route. And seal it for another couple of inches either side, making sure surfaces are clean and dry, obviously. That said, replace the rad. alone, with a brand new one, and you haven't really fixed the problem - have you?

Why the aluminium foil, I don't know. Any hard-setting gunge is going to work, given time to set properly. So, in answer to the actual question - yes, but maybe forget the foil, and either carry on, and pray, take it to a mechanic, or seek the potential source of why it needed to split, yourself.