[old_tv_nut]

Your tube tester may be telling you whether the tube's emission is normal or weak, but it will tell nothing about the frequency response.

What do you mean - "need to let it run a good amount of time?"

The difference in response with different tubes is way too much. It looks like the last waveform you posted is grossly mistuned, the dip at the 4th major division from the left should not be there, just a smooth slope like the next to last waveform.

From your chart it looks like you replaced the few caps that were grossly out of tolerance., so unless you made a mistake resoldering them, there's no clue there.

Do you have more 6EJ7s to try?

Could we be missing something that can't be seen in your posting, like a dirty socket? What happens if you wiggle the tube?
Is the tube shielded? Does the shield have good ground contact?