I have several large areas of gravel in my yard (put down by previous owner).. and I find extended control glyphsophate the best method for weed control. I tried organic methods like pulling by hand, fire torch, etc.. nothing worked. Weeds broke off easily because they were growing through membrane and had a good grip.. the fire just burned the tops and the weeds became multiheaded hydra instead, and I couldn't keep on top of the mess. I was spending all my time battling instead of weeding my veg beds and other areas that needed attention. The soil under the rocks and membrane is rather sterile as it is, so I gave up and spray weed killer now. The extended control is pretty good, and means I spray far less often (reducing my exposure time).
As others said, getting rid of it and planting up with shrubs is ideal.. but if you like the gravel area.. spraying is probabaly your only realistic option in these circumstances.
I have gravel area to, if you want to start from scratch then above recommendation is good offer it for free let the bulk go, then tred in the remaining gravel and lay membrane on top and re gravel sorted.
To answer your original question - I can't see why you shouldn't lay new membrane on top of a layer of gravel, as long as the gravel doesn't puncture the membrane. If the remaining gravel is angular, that might be a problem - you could add a few bags of sand, I suppose, to even it out first... and/or a layer of that blanket stuff you put under pond liners - and make sure you really have removed all the weeds, maybe spraying with path weedkiller before putting the membrane & new gravel down.
Since 2019 I've lived in east Clare, in the west of Ireland.
It's all about how heavy things are and how far you have to move them. If you're methodical about it, you should be able to replace the membrane, rinse the existing gravel and put everything back as was. Either that or a mini digger, a couple of skips, and a pile of money
Once you have it properly under control it shouldn't take long each year to keep it looking OK. As you have already removed a fair quantity I would be inclined to level it, lay a heavy duty weed fabric and then lay the remaining gravel back on top. Although weeds will appear it will be a simple job to spray it with week killer about once a month during the growing season. That shouldn't take more than about 10 minutes including the time to mix the solution and pressurise the sprayer.
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As others said, getting rid of it and planting up with shrubs is ideal.. but if you like the gravel area.. spraying is probabaly your only realistic option in these circumstances.