Public transport is far, far more cost effective than car infrastructure. And that's just direct costs - not even including the cost of sprawl (which makes all other infrastructure more expensive), road deaths and injuries, noise, pollution, storage costs for vehicles, the health costs of inactivity and social isolation, etc etc.
> build good transit for the busiest part of the city (downtown) and build large parking lots around the terminals
This is a terrible idea because the numbers simply do not stack up. A typical metro train can carry roughly 1000 people. A large car park might fill half of a single train. At a station with good frequency, a train will leave the station roughly every 5 minutes.
A much better idea is to run good regular public transport to the station, build bike paths to the station and quality bike parking at the station, and build more housing at/near the station instead of a big parking lot.