You can still build one. I still can’t – in any reasonable way – poach whichever oil rig workers you choose to underpay.

Lets breakdown what those costs are though.

1. Some portion is paid to workers to construct the oil rig.
2. Some may be paid to the government as part of a lease agreement generally to compensate the public for your exclusive use of the well.
3. Some is paid to previous suppliers.

3 is really a combination of 1 & 2 so lets exclude that. For 2 we could have a government that takes this money later. Often this is the case for a lot of these deals. A lease you can pay later and royalties are paid when you actually produce product. That really just leaves 3. If you were able to compel these workers to work for you without compensation then this "How does capitalism empower people to own what they produce?"wouldn't be true so you'd have to offer some compensation but that compensation could be equity in the form of a workers cooperative.

Is it more difficult for you to compete, sure but that's like saying it's more difficult for me to be an artist. I think we should be talking about where we are stepping on the scales for one or another.

United States tax dollars, in the form of DARPA grants, paid for the development of the internet. So there is precedent for extremely expensive operations to be successfully carried out under democratic control

You could government fund everything if you want. It's usually quite beneficial in things you suggest which are early stage how it would be commercial viable is pretty uncertain. But there are trade offs.

• In market economies you may need to raise the money with debt. If you are an oil producing economy and suddenly the oil price drops you may no longer be able to pay those debts. If you instead lease it to a private company which you then collect royalties or taxes from you don't have to take that risk. You could fund it with taxes however that limits your growth if you are a smaller economy.
• With the right incentives companies will compete if you have just a single nationalised producer where bureaucrats don't the same level of incentive as owners they likely will run it less efficient. There is obviously the case though that often private companies push for regulations that limit competition and try to reduce their costs for externalities they impose on others.

Also, since oil deposits are a natural resource, one could argue government ought to be involved in their collection.

I think it's correct to say government should be collecting revenue from the natural resource but I don't think they need to specifically be the ones running it.

I should clarify I think capitalism is great but doesn't mean our implementation is perfect (and an example is privately owned land).