Actually urea isn't extracted from petroleum per se, rather natural gas is cracked to make hydrogen, which is reacted with nitrogen in the atmosphere to make ammonia, which is then reacted with CO2 (which is a byproduct of the previous step) to make urea.
This process could actually be greened up quite a bit by making hydrogen by electrolysis with renewable electricity rather than cracking natural gas and using waste CO2 from some other industry, like a power plant. Doing that would make the fertilizer business carbon neutral and sustainable.
You're right though in that plants don't care where the urea comes from. I consider organic gardening to be mostly a matter of caring for the soil with the understanding that healthy plants grow from healthy soil. Improving the water retention and root spread with lots of organic matter in the soil allows the plants better access to whatever type of fertilizer you apply, and of course tilling in compost provides food so you may not need to fertilize. Planting densely and using cover crops keep the soil aerated with root activity, capture nutrients that would otherwise leech out, and conserve water.
Other parts of organic gardening are more voodoo than anything else though. Some parts are even bad, the strip mining of guano based phosphate on tropical islands has increased because that phosphate is 'organic' resulting in increased demand for the stuff. Kelp meal is also a common organic ingredient, but kelp only grows in several acre patches within a couple hundred yards of shore along the coasts. It's a pretty limited resource despite it's fast growth rate, and the kelp forests are the coral reefs of temperate waters - with the kelp at the base of the food chain. Marine protected areas and increasingly stringent regulations are reducing the supply of the stuff. Peat is also being used unsustainably, and shipping this bulky stuff all over uses lots of fuel.
Organic fertilizers do contain other things than just NPK, including a host of micronutrients, which aid their effectiveness, particularly in potted plants where micros are more commonly a problem. There are conventional alternatives to that though, such as epsom salt (Mg) and liquid micronutrient solutions such as floramicro. I've put those products in aquariums with tropical fish with no adverse affects - they won't hurt your soil. You'd still be better off topdressing with compost, cultivating shallowly and watering it in because that will improve the soil as well as feeding the plants.
I'd really like to see people get away from 'organic' and more toward 'sustainable'. Sustainability can be analyzed and measured objectively and is important to society in the long term, where 'organic' is just a jumbled list of arbitrary rules based on marketing to peoples fears (GMO, chemicals, big business) and misconceptions (natural=safe/good) to justify higher prices and/or exclusive customers. There's nothing wrong with that, but we don't have to be orthodox organic to market our tomatoes to the wife and kids.
The best time to sow is 5 years ago.