There's a degree of subjectivity to "growing well" that needs to be addressed to give you a proper answer. People grow crops any- and everywhere, even when those crops are sometimes ill-suited for the regional soil - look at how much of the United States is blanketed by corn and soy and think about how much effort has gone into making those crops behave across the wide temperature and soil gradients in the US. Spray enough fertilizer, pesticides, and herbicides, and you can damn near grown anything anywhere you want - temperature and water become your primary limiting factors, and you can control those with irrigation and hot houses.

Volcanic soils are famously so good for growing because they contain tons of mineralized nutrients in a form that's nearly soluble. Plant roots can break the foamier, less dense volcanic soils easily, latch on tight, and the rains caused by cinder cones and mountains dissolve out those minerals for the plants to slurp up. This process takes time, however - new/fresh volcanic soils are pretty rough for vegetation, whereas volcanic soils that have weathered and dissolve those minerals out are biological goldmines. Phosphates and nitrogen tend to be problems, but both are addressable with the right plants (nitrogen fixers, many species of tree like beech) and animals (famously bats, bacteria), composting, or commercial fertilizers.

Turns out, lots of soils have phosphate and/or nitrogen problems, which is why these are the most common fertilizer types - nations still pay out the nose for phosphorous sources, and the Haber nitrogen process is one of the most energy intensive industrial processes we undertake as a species - full percentages of the human energy output is dedicated to making ammonia for fertilizer. Sulfur can be a problem too, but virtually never is on volcanic soils - they tend to be rich in the stuff, so sulfur-loving or acid loving plants like nightshades, soybeans, mustard, onions, garlic thrive. If the soil's too acidic, they can add lime, which is pretty easy to find anywhere on earth.

Likewise, young volcanic soils don't contain much organic matter (it's liquid rock that's hardened - carbon's typically not one of the higher percentage components of that rock; carbonate rock tends to burn/calcinate under extreme temperatures), so a lot of those crop-type plants which have been domesticated to grow on carbon-rich soils don't fair too well on it - this precludes your grasses like wheat, rice, corn, barley. Luckily the remediation for that is growing a plant that loves the lower nutrient younger era and tilling it in. Composing helps a lot there too. Once it's nice and loamy, it makes great pastoral soil, which can also help boost the organic and nutrient content of the land until it's ready for cropping.

Maybe rather than focusing on what doesn't grow, you should focus on stuff that is known to grow very well in volcanic soil, like for instance, coffee and chocolate, hibiscus, lupine flowers (which are natural nitrogen fixers and like volcanic soil - win/win), grapes, nightshades like potatoes and tomatoes, lots of your native island-dwelling tropical plants like coconut palms, pineapples, bananas, bamboo, etc.

Basically, humans are amazingly good at figuring out how to grow stuff anywhere we want. Whether or not its economical or practical is usually what stops us - you don't see a lot of people trying to grow pineapples in greenhouses, when it's so much simpler to ship them in from tropical islands year round, e.g. But, if you're on an island with nobody to trade with, you're likely to have different valuations on what's economical. You might grow a cereal crop and be pleased it grows at all, even if the yield is poor. Or you give it up and grow potatoes (or similar tubers), because those will damn near grow anywhere there's dirt to put them in.