I believe there are six ways you can go when you break or strip a screw:
- screw extractors
- left-handed drill bits
- do something to the top of the broken screw that would allow you to extract it conventionally
- plug cutter
- drill adjacent holes to allow the screw stub to be grabbed by pliers and turned
- ignore the broken screw(s) and re-site the relevant fixing
Sometimes the choice of what to do is based on expedience, sometimes it's based on the desire not to damage the surrounding area (screws can be mounted on show surfaces) but thankfully this doesn't apply to you so your options are open.
Screw extractors and left-handed drill bits rely on a simple principle, that their left-hand threading matches the withdrawal direction for normal screws so driving them in acts to unscrew a stripped/broken screw.
Option 1 I think is out here, unless the screws are particularly beefy (which I'm guessing they aren't otherwise their heads wouldn't have sheared clean off!) as IME screw extractors are only really viable if the head is stripped. Once you break that off there's just not enough steel left for screw extractors to work well, or at all — the head of the extractor has to go into a drilled hole in the stub of screw1. So for direct extraction that leaves option 2, left-handed drill bits, and you have to drill a hole anyway.... I've never used these but seen some demos online and they can work exactly like you'd want, but I think it's inevitable that this will vary.
3 nearly always results in damage to the surrounding surface, but since this is a bed bracket this may not be a big deal here. One of the simplest versions of this technique involves cutting a slot in the top of the screw using a cutting disk mounted in a Dremel-type drill. The screw can then be extracted using a normal flat-head screwdriver. This works well for screws that aren't stuck, but the longer the screw is the less chance you have of it working, and ditto if the screws have rusted (even a little) or the wood just has a particularly good hold on them (often the case with hardwoods).
Now to option 4 which is pretty self-explanatory, you drill out a plug of wood that contains the broken screw2. This works great usually, but a largish repair is needed afterward. So while you're virtually guaranteed to be able to get the screw out it's more work overall so it is reasonable that this often isn't the first thing many try, but it always remains a great fallback option. After removing the plug you can repair the hole with a good filler3 but it's considered best practice to fill the holes with wood, so you glue in a dowel or face-grain plug4. Then after the glue has fully dried you drill new pilot holes and install new screws.
5 is crude but can be effective. It's best to use fine-tipped or needle-nose pliers but even so it makes a mess of the wood surface that requires a fair bit of repair work. But where the surface won't be seen (as here) this is certainly an option worth considering and easier to manage than cutting out a plug of wood. Also far cheaper and faster if those are important factors.
Option 6 I would presume isn't viable here as you have three other mounting locations that you need to align to, and repositioning all four probably isn't possible.
One additional point re. this bed
I would take the opportunity to remove and replace the matching screws on the other bracket(s) as they may be defective or unsuited to the load requirements.
1 Which is ideally perfectly centred, not always easy to do. So in addition to the screw extractors themselves you have to have some appropriate drill bits and you really need a centre punch too. If you don't have any of these already this is a lot to buy just for a few stripped screws although of course they all have potential future uses.
2 Take out a matching screw from another bracket to determine the depth of plug you need to cut.
3 Some fillers are strong enough to accept the replacement screws, e.g. Durham's water putty and epoxies (homemeade or commercial) both of which can be coloured to more closely match the surrounding wood. Note that on a show surface putty will always be visible, no matter how good the colour match.
4 Face-grain plugs are normally to be preferred if possible as they are both aesthetically more pleasing and provide a firmer anchor for screws (see this Q&A for more on this). I mention this for completeness and for the benefit of future searchers whose requirements may be different, here the aesthetics are irrelevant (surface hidden by bracket) and the strength advantage of face-grain plugs over dowels may not be needed because the force exerted will be largely at 90° to the screws.