CALLING ENGINE GUYS/GALS

I'm thinking the OP had some serious suspicions about the whole deal to partially tear down a refreshed engine on the basis of a bad leak down. The air coming out of the intake would have concerned me if I knew for definite that the valves were closed. It's usually not a bad idea to double check a refresh if it's been in storage a while anyway. Especially with the replacement cost of a 340 block if it gets a window put in it from a rod coming out.
Finally, someone mentioned "Valves". I agree with most here, that this motor needs a complete go-over, including an over bore. The intake pop has nothing to do with the block, heads yes!!!. Were the valves lapped and the valve stems checked for wear? Were the "J" heads cut for the 2.02's done correctly (they look like two valves may touch. I spent over $4,000.00 on a complete rebuild on my 340, 35 years ago and it was money well spent. Note: A motor should be built for its use. In my case, I built a tight clearance block for long use, where a loose block clearance would be for a race app. There are so many variables, including oversized carburetion, cams, torque converter, rear gearing and tire size, that should be considered in an engine rebuild.