The typical fix I read about is wood glue, etc. But I am concerned it won’t be strong enough.
This is a valid concern and sub-strength fixes are common in these sorts of situations.
From the Comments:
it would be hard to get enough glue in there to achieve the desired adhesion.
Yes, in particular it's difficult to get glue deeply into a closed crack to set yourself up for a thorough bond. But there's another important factor and that's how clean the surfaces are inside. Old cracks on old furniture can be really filthy! But even with a crack on a recent piece the wood surfaces have aged, and only fresh wood surfaces bond very strongly, see footnote 1.
As a result breaking off the split piece can be advisable as it gives full access to the mating surfaces to clean them. This is desirable in every case but absolutely vital if a prior repair was done and old glue remains in the crack, and if it can be managed without doing excess damage. Some judgement is needed here but it remains a *gulp* moment even when you've done it a few times.
If you can break the piece off
Since sanding the wood surfaces isn't an option at least brush the surfaces down with a solvent such as denatured alcohol (UK: methylated spirits) acetone or lacquer thinners. I generally do this twice, because why not?
Carefully test-fit the two pieces together (try not to handle the mating surfaces) and see if there are any bent splinters that are going to prevent the wood coming together properly. If you find any it's generally best to just break those off rather than try to align them. DAMHIK.
If you can't break the piece off
If splitting off a piece is not an option there are still some worthwhile tricks to know. Clean and blow out any dirt or dust from the parts of the crack you can see. You can temporarily lever the crack open to help with this but try not to dent the wood with your levering tool whatever it is.
Now to getting glue in the crack, first off don't skimp on it. You're going to make a mess so be prepared for this, and you will be using an excess to try to make sure that glue has gotten to where you want it to be. Where possible this means glue will drip from the bottom or underside of a crack, it also means that you will apply it to the surface or areas you can access more than once during the operation as the glue disappears inside. Work the crack open and closed a few times to try to get capillary action to carry the glue along and downwards, then apply more glue and work it in from the top. You see another reason that superglue isn't a good choice here.
Glue can be blown more deeply into a crack by blowing through a straw or using an air hose, either can be surprisingly effective and sometimes will get glue through to the other side. Glue can also be worked into the crack with thin tools2. With a crack that's open at one end you can also use strong thread or dental floss (unwaxed if possible) to carry glue in, using a sort of sawing action.
After glue, clamp time
If using epoxy heavy clamping pressure is not needed which is another advantage with epoxy. Epoxies are also excellent gap-fillers making them the choice for damage repair in many cases, i.e. where there's missing wood or a crack that won't close all the way. You do want to squeeze the joint shut of course but there's no need to really mash the wood together, which basically you do want to do with PVA.
With PVA you must apply firm clamp pressure to achieve a strong bond, so don't skimp on tightening them. My standard is that you have to use pads of scrap wood to protect the workpiece surface from denting (this also helps spread clamp pressure, which is usually desirable).
After the clamps are on and you've wiped up all the glue squeeze-out put the chest off to one side for the glue to dry/set and don't disturb it for any reason for a good couple of hours. Wait overnight before taking the clamps off if you can (regardless of the stated setting time if using epoxy).
If in doubt, reinforce
We can't know how your fix is going to go. If you feel it's needed there's no harm in reinforcing the repair somehow. This can be done with screws or dowels across the joint, my preference would be to use dowels as they can be more easily driven in dovetail-fashion (at opposing angles) which directly counter any tendency of a joint to open up. Hardwood dowels are to be preferred, and 1/4" (6mm) is the max diameter I'd use myself.
1 What do I need to do to prepare wood for gluing?
2 Small palette knives, the small blade of a penknife, stir-sticks 'borrowed' from your favourite coffee place and bamboo skewers can all be utilised here.