Start small scale and build.
You're right about today's industrial scale mining operations requiring massive amounts of infrastructure to support, not to mention the energy cost. We invest in it because of the net benefit in access to material resources, but mining is not something that (today) a small investor can get into by him or herself. In order to be efficient and return your investment, it needs to be large scale right off the bat.
Your world however introduces a new dynamic to that. Your massive tip truck MAY cost around 50m, but shipping it to the new world now makes it cost $500m, and that doesn't cover the cost of parts, fuel, maintenance personnel, etc. In other words;
1) Either you invest 10x what it would cost to develop a large scale operation on Earth, or
2) You invest the same amount, knowing that your investment will take longer to realise benefits.
So, you put together a small team of miners, complete with enough infrastructure and plant to built a small mine, and a small smelter. The single biggest issue here is energy and food, but I'm going to assume here that when you say shirtsleeves planet, you're implying that there's life there, including food or the capacity to farm it in that soil.
So; you send over as many solar panels, batteries, wind turbines as you can to power all this. You get your mines (and your farms) established, and you smelt what you get out of the ground. That gives you metals, but you don't send them back to Earth. You reinvest them in more mining equipment, more smelters, more energy systems. In the meantime, your farmers and miners are having kids, which means more workers. Your industry grows, making more use of the more materials you harvest, etc.
In the meantime, Earth has still been depleting its mineral resources, driving up prices. R&D in space travel continues, bringing down the price (even slightly).
Eventually you reach a tipping point. Your planet is producing more materials than it needs, Earth has stronger demand than ever and shipping costs come down to a point where it's worth it to ship the minerals to an Earth market. In the meantime, your colonists want some of the finery and sophistication only possible from Earth, and a trade agreement is reached. Your planet is effectively a free state willing and able to trade with Earth.
The catch? Your initial investment may take centuries to pay off. In that time, the colony could die off from unforeseen circumstances on the planet, Earth could wipe itself out in a war, Once the colony is on its feet it might not want anything to do with Earth - any one of hundreds of possibilities may manifest which could wash away your investment. Even if they don't, it's your grand-kids (or later generations) which get the benefit of your foresight, not you.
But, if you want to start small, this is the way to do it.
How small is small? Think a mining town, and a steel town rolled into one. You need farmers, carpenters/builders, miners, refiners, chemists, prospectors, doctors, engineers, heavy equipment operators, road builders, entertainers (which probably includes brewers) - everything you need in a regional town and the surrounding environs that is capable of making an area self sufficient.
This is an estimate, but I'd imagine you'd want to start off with no fewer than 50k people (staged obviously; you want your builders and farmers out there before the miners) and all the supporting seed and live stock, starting food supplies, building materials, energy production plant, etc.
But with that, you can build a self sufficient community that reinvests in itself and grows to a point where it's capable of trading with Earth several generations later.