You really think it would be a full time job? Yes. You may not need to be on site while each of the trades is there, but you will need to stop in from time to time to make sure they are doing things right and staying on schedule, or adjust the schedule if they are running behind. You will also need to spend time ordering materials, and scheduling the trades and inspections.

I'd have a plumber and electrician bring in utilities (water, gas, sewer, electrical) from the street.

Often times it is the utility company that brings services to the house, but each utility is different. Check with the utility company in your area to see how far they bring utilities, and utilities are often one of the last things to get connected.

Then I'm curious where I'd go from there.

If you have to ask this question, don't be your own GC, or you need to do a LOT of research. Once the house is framed, your next priority is to make it weather resistant as quickly as possible, house wrap, shingles and windows then your finish material (stone, brick, siding).

I could then spend the next 6-8 months roughing in the plumbing, electrical and HVAC

Check with your local code officials as each jurisdiction is different. Many will not let unlicensed individuals pull permits, so your HVAC/Electrician/plumber may need to be involved.

Keep in mind that many jurisdictions (and lending companies) have expiration dates for their building permits. They want you to finish the project as quickly as possible, and if you don't finish in time, you may need to pull all new permits. Lenders may be hesitant to lend money for new construction projects, and even more hesitant if you are not using a GC. Your local code official will not give you your use and occupancy permit until the yard is sodded/seeded and all areas are finished (excluding spaces that were identified as "unfinished" on the plans.

To give you an idea, when I built my custom 4200 sf house, it took about 1 year from the date we signed the contract until we had the U&O permit.

The lot next to ours is going through a similar process, but the owners are acting as their own GC. They are currently at about 22 months into their build and they were just hanging drywall last week. Today, they had the trim contractor in, and had the utility company in to replace their temporary electric service with the final connection, but they did not realize that when this happens, the utility disconnects power until an electrical inspection can be done, the result, the house will be without power for a week, and no progress can be made unless the owner brings in a generator so the contractors can have electricity and light.