My JF Riders vol # 8 lists several notations for "intermittent reception" on the Zenith chassis mod 1004.
If you are using the orig tubes since powerup w/ variac, recheck them to make sure 1 or more may or may not be breaking-down after warmup. Internal tube-shorts etc? Recheck your socket-pin connections to be tight & clean.
Your Zenith chassis 1004 has a old orig style Zenith factory "candohm" type wirewound resistor mounted to chassis. These parts are most usually bad always, intermittent in themselves, and generally overlooked when recapping a set. Check it closely for intermittent operation.
Heres the probs listed by JF Riders vol # 8 (published in 1937)to search for "intermittent reception" probs on the Zenith 1004 chassis:
(1) tubes ( recheck em under load, replace them one at a time if you have spares)
(2) IF trimmer short
(3) dirty variable condenser plates ( can wash-away yrs of debris with spray-can-type of electronics cleaners: "Blue-Shower" or/ "Big Bath" sold at select electronics components stores.
(4) poor ground at "candohm" resistor
(5) loose link wire across Z & G on aerial strip ( check your antenna wires-terminals closely for shorting across these 2 screws)
(6) poor contact on bandswitch. ( see #3 suggestion for electronics spray cleaners) to clean switch contacts. (rotate that switch and make sure contacts are not intermittent from age & debris buildup and it makes secure contacts when engaged)
(7) defective aerial (shorting somewhere internally in the loop, or broken connection internally?) wiggle it when radio is powered up.
(8) defective bypass condenser ( not likely if you already replaced "ALL" them?) Make sure you didnt miss any, I personally dont find many bad "new" caps in my stock, but it can happen.
If your old "candohm" is intermittent, replace it with a close approx suitable ohms-rating, in the 5-10 watt ceramic type wirewound power resistor type(s) using terminal strip(s) in place of the old candohm metal mount assembly. Many of these are "sectional", so you will need a schematic for the data, if you dont have one, I can help provide the data.
Best of luck with your Zenith restoration!!
Sincerely, Randal @ TxVintageRadios