I've had similar problems to yours but not exactly the same. I've "done" several of these engines over the last 45 years or so. The ones with the adjustable lifters have sometimes been noisy. The quietest engines have been the ones with the original style lifters. I've often questioned whether the solid style adjustables are as good as the hollow type. I especially worry about the weight difference. I've re-surfaced original style lifters completely flat on a valve grinder and have had good results, although technically they are supposed to have a slight radius on the base.

I have had engines with adjustables that were whisper quiet. The last two engines I've done have had mild "clicking" on one or two tappets during the warm up period, and are then very quiet when the engine is completely warmed up. The quietest engines were ones that had reasonably low total mileage on them and the noisy ones were probably very high or unknown mileage blocks. I know that these "symptoms" vary from yours, but here are a few possible clues:

Was the camshaft a replacement? It may have been reground without the correct clearance ramps. Is the camshaft or lifter bases worn excessively? Did the rebuilder return all components to their original positions if the original cam and lifters were reused? Not all flatheads have the same valve clearance specifications, and parts may have been mis-matched. 1951 valves, springs, and retainers are different and may have been replaced with incorrect parts.
A local engine machinist with 40 years experience has told me that the lifer bores and/or valve guide bores are very often out of spec. and out of alignment with each other on these blocks, and it is necessary to resize and align them with each other to gain a perfectly quiet engine. I believe this to be the problem with the 268 cu. in. engine that I have in my F-1 4x4 pickup right now. Just a few things to think about.

I bought a used rebuilt Mercury engine for my Ford several years ago that had all the symptoms that you describe. I changed out the cam and lifters, set the valves to proper specs, and had no further noise problems. At the same time, I re-surfaced the valves and made sure the valve seats were concentric with the valve guide bores. This is very important, and without the proper tools, the valve seats can be "messed" up such that they are not in line with the vavle guides. This is bad news!

I've heard this same noise when hearing expensive show cars start up their engines, and I know the frustration they must feel. Some of the owners may not even know what a flathead is supposed to sound like. There are fewer and fewer of us left that know what these engines sounded like when new. If anything, you might hear the sound of the diapragm moving in the fuel pump, or the "whir" of the geneator brushes!

Good luck!