I’m an architect.
Assuming you could find exactly what you what (site, design that fits, building quality, efficiency, etc). It isn’t likely that you will do better than buying the existing house. But it is unlikely that you would find a perfect situation.
In my location, the going rate for architects to design a custom house is 15% of the construction cost. I won’t do a custom house for less than 20%.
What you should be getting from an architect is:
Design that fits your family.
Design that fits your budget (assuming your budget is reasonable for your location) and meets the jurisdictional requiremenets (building codes, development by-laws, etc.)
Contract Documents (drawings, specifications, bid form)
Architect will issue tender documents, evaluate bids and review with you.
Administer the construction, visiting the site as required, to ensure that the house conforms to the design you approved; review contractor’s monthly draws to ensure you aren’t overpaying, and ensure that your house is lien free prior to final pay-out.
@boffin’s suggestion (do your own interior finishing in all, most, or some of the house) is good for saving money.
If you have no experience in construction projects, an architect will be your safest choice for a problem free building experience.
You do have to go through a reasonable interview process to ensure the architect is one you are comfortable with.
If you intend to build a house from plans you buy from a plan service, be careful of the plan service. Many don’t provide enough detail to prevent a contractor from making substitutions that may be bad for you.
In my 30+ years of experience, I have only encountered 1 contractor who didn’t intend to build what was bid (low contractor, government financed, project). But I have seen the fairly widespread disasters that contractor’s can inflict on their customers.