It does indeed sound like your bearings are shot. In some cases a worn bearing (if it's very worn) can cause the stator (non-rotating part of the motor/dynamo) to start rubbing against the rotor (the force the tire exerts on the dynamo pushes the rotor/rotor axle out of alignment causing it to rub against the stator in very bad cases. This would increase the amount of sound more than just a worn bearing.
Note: in some cases a slide bearing/plain bearing/teflon bearing will be used in which case you can't easily service it when it's worn, you could look for replacement parts but it's unlikely they will be available. Bearing type I am referring to can be seen in this image:
From what I've read so far there are two types of bottle dynamo design, one where the stator core (soft iron core) is positioned around the rotor such as in this image:
Disassembly of this type can be seen in this video:
And another type where the winding of the stator and the core are positioned around the rotor, as can be seen in this video:
I am not quite sure what type your AXA HR dynamo is but in practice it doesn't matter much.
They appear to be relatively easy to disassemble judging from the disassembly videos posted above. You could firstly check for play prior to disassembly as suggested by @carel . After disassembling (if there is play/the bearing feels bad) you can take out the bearing, clean the balls/cage(if present), clean the bearing races, apply new/clean grease. And reassemble. You could theoretically replace the bearing but you'd most likely be better off buying a new dynamo when considering the part/labor cost.
I could not easily find bearing replacements specifically for the HR Traction but if you measure the ball diameter you can find those ball bearings seperately and replace them, but do keep in mind the bearing races will probably also be worn and these are in general non replaceable parts. So you will most likely not get it as good as new by replacing only the balls. Tool used for measuring ball bearing balls: You can also use a caliper or micrometer if that's all you have.
What you could perhaps also try is to loosen the nut that secures the cap of the dynamo a bit, if that is over tightened than I believe your bearing will also be too tight (just as a headset bearing can be too tight and will cause the bearing to not run smooth at all). Try loosening it a bit and see if it runs more smoothly/quietly afterwards.
Also make sure to clean the dynamo properly, even though sand/grid getting under the cap or even in the dynamo is quite unlikely, it could be considered a cause of excess noise depending on how the bike is used (in muddy conditions I could imagine sand getting under the dynamo cap).
In my experience it is quite normal for bottle dynamos to make some noise, you will definitely know when they are engaged due to the sound they produce. So they won't be fully silent, for that you'd need a hub dynamo.
There are many alternatives to axa HR bottle dynamos, they are quite common in the Netherlands: here are a few: https://www.internet-bikes.com/zoeken/11506779-11506787/?search=dynamo