Question about cold frames.

I live in Brooklyn, NYC. I want to start some plants in flats as soon as possible but I don't really have any indoor space to do that.

My thought was to build a cold frame and plant outside earlier rather than later.

Most of the instructional videos I've seen only deal with using a cold frame on an area of land where the plants are seeded directly into the ground, OR in a bed that they will not be moved from.

I'm wondering how to build a cold frame specifically for starting seeds in flats that will then be transplanted into the ground. Any other things to consider?

What would this setup look like? How much warmth will the cold frame provide and how early can I start those seeds?

It's still going to get cold at night, probably too cold for those southern crops to start well because they want a steady temperature. You can help by putting mass inside to warm up in the daytime. Bricks, a wall of your house, patio stones, rocks, bottles of water etc . Consider a a blanket at night. Maybe a small heater, old waterbed heaters work pretty good and have a thermostat, even an incandescent light bulb will make a fair amount of heat..
It's going to get hot in the sunshine too, and quickly so you'll need to be around to open it up a bit to regulate the temperature.

I see no design difference between a cold frame that holds containers vs planting in the ground.

I would advise to bury the containers flush with the ground. Sitting on top,  the pots and roots will freeze quicker.

Here's one I built recently. It functions well but I'm down.south.