Best way to clear gravel areas of weeds on allotment - weed killer, or muscle!?

Depends on what you mean by best.    Weedkiller should see the stuff off in about 3 weeks and you can then just lift out the dead stuff or put a hoe across it.

Personally I think that takes too long and save the weedkiller for annoying weeds which don't understand the word no. Take a rake, scrape back the gravel (looks like bark in the picture) and just dig the weeds out. Looking at what you have there it shouldn't take very long at all.

Search for some plot pictures on here and you'll see how lucky you've been with the plot.

If putting gravel down it would be better if it was on a geotextile fabric (such as Terram) which will let water drain but stop weeds from coming up.  So perhaps consider lifting gravel and putting fabric under it once you've conquorered the weeds this-time-around.

But, for me, the plot is far too much path and far too little growing space.  Sorry! So I would be inclined to rearrange all  the raised beds to provide more production area.

If you use a weed killer based on Glyphosate it won't poison the soil, and it only kills plants that it comes into contact with - so you do need to be careful not to allow the chemical, including any spray-drift [so avoid a windy day] to come into contact with anything you want to keep. You also need to apply only when there will then be 6 hours of dry weather.

Opinions vary on whether to use weedkiller or not, and whether folk think that companies like Monsanto are monsters so its a choice for the individual.  Personally I would use a Glyphosate based weedkiller for a one-time "flying start" every time.  If the weeds get away from you then you will spend all Summer try to sort it out, wind up miserable at the effort and possible "give up" and, for me, a Newbie giving up is the one thing I would like to try to help avoid

I can see the appeal of a complete redesign but if it was me wanting to get a few beds going I  think I'd get a lot of quick progress against those weeds with a hoe whilst thinking about how to use the extra space.

In my garden where there is pea shingle type gravel the weeds come up really easily especially if you really wet them first.

Good luck with the plot - I like the rockery - looks a bit like a herb spiral.