The articles I linked expand on this and there are many resources to research this.

Short of cutting the tree down yourself, hauling it yourself, and then processing & building it:

The C02 cost of harvesting, hauling, and processing trees into wood is very costly.

Keep in mind that the current alternative to building with wood is to use concrete, which has a tremendous CO2 cost [1].

[1] https://www.bbc.com/news/science-environment-46455844

We should also look to compare hempcrete or similar lightweight materials with superior insulation qualities; "an un-rendered 30 cm thick hemp concrete wall enables a storage of 36.08 kg of CO2 per m2 [1]."

[1] https://urbannext.net/hemp-concrete/

But is that any different for any other material like steel or concrete?

If we can get all those steps onto electric power instead of internal combustion, then (assuming the power's coming from renewables, hydroelectric, or nuclear) that should eliminate the issue, right?

I agree completely. I live in a century home. The spruce lumber in my walls is safe and dry and has been there decades longer than it would have been if those trees had been left to die and rot on their own.