Yes, but...
... we use industrial processes for a reason. Production, recycling, and manufacture boil down into "synthesis" and "extraction", and are different areas, so we'll examine them individually. Ultimately, it's best to have both playing to their strengths.
Synthesising (from provided material)
Let's look at making a conventional CPU. It's a slice of ultra-pure silicon with an aligned crystal structure, plus some trace element dopings. They're made in clean rooms for a reason. That's then etched at nanometre scale. (For comparison, small cells are 100 times that - you want something about the same scale as individual protiens.) This one is a clear winner for the industrial side.
Now let's look at making a wooden chair. This is much easier, it's just a (very) odd-shaped tree ... doable with the kind of genetic engineering you'd be after. You could grow the biomass in a few days or weeks, so it's a question of getting the shape right and dehydrating it afterward to prevent decay. It'll be slower than industrial processing, but suns on in-situ solar and self-assembles from atmospheric carbon. This one is borderline - it it worth re-engineering things for every redesign? But chairs don't change much.
Food production is easier again, and algae/insect/bacterial/fungal production is being researched today. (Yeasts are perhaps the first form of that, and have been in use for "a long time".) Pharmaceuticals are produced this way also. This one's a winner for "nature, with a controlled environment.
Extraction (from waste or wild)
This is what the phytomining techniques you refer to are. Let's suppose that you want to mine nickel. Ores are generally under 2% nickel, which is also considered the critical concentration for viable farming-for-metal today. So plants can concentrate soil metals, often from contaminated areas.
The thing is ... plants don't grow pure-metal logs, they just have oddly-coloured sap. To extract metal ingots (or whatever form you choose), you run them through an industrial processing chain. If you have one of those on-hand, you probably will mine raw ore, too, rather than waiting for small volumes of plant to grow on the surface. The dust from that ore and the waste from your extraction process can become the soil for your crop.