Not only can't you do consulting work for your foreign connections while you're in the US before you've obtained your employment authorization document (EAD), you can't obtain an EAD to set yourself up as a sole proprietor consultant. The process of getting an EAD is first to obtain an offer of employment in the US and then to apply for the EAD.
The office processing your application depends on which international organization your wife works for. It will be either the Office of Foreign Missions (which says that applications normally require 6 to 10 weeks but in August 2021 was saying that they took up to 16 weeks) or the US Mission to the UN (which doesn't say how long they take).
US mission link:
I'm having a hard time finding the actual conditions of the G4 visa
The terms of your status under the G-4 visa depend on the agreement between the US and your spouse's organization. For example, if she is employed by the UN, they depend on the UN headquarters agreement. These agreements are broadly similar, of course, but the fact that the status is not governed by a single legal instrument is part of the reason for your difficulty.
What is defined as work WRT G-4 spousal visa?
The problem here is really that the US avoids making clear statements answering that question with respect to nonimmigrants in general. The same question is frequently a problem for tourists, business visitors, students, and so on. There are some questions that have been answered clearly. For example, "volunteering" to do a job in the US that is normally paid is not allowed because it still deprives a person who is authorized to work in the US of employment (and it could be a cover for labor abuse). This objection doesn't apply to doing remote work for a foreign employer, of course, and such work is generally allowed in some jurisdictions, including Canada. US law, however, like that of most jurisdictions, hasn't accepted this way of thinking, and the US government hasn't shown any signs of wanting to.