Your voltages look fine. I wouldn't use the word 'failure', these kits are supposed to be fun and teach. Wow 64 page instructions that covers a lot. Common mistakes are mixing up the transformers (red, yellow, white, black) some IF transformers have a capacitor inside them, which may short to solder blobs.
I believe it uses negative AGC meaning the TP5 AGC voltage goes down with increasing signal in order to lower Q8's gain.
With the local oscillator disabled C29 to GND, there should be no AC signal downstream at TP5,4,3 and AGC voltage should go up to the highest possible ~1.4V at TP5 or TP2, and you get 1.44V which is ok.
If the AGC voltage is still low 0.7V and there's AC signal present, then the IF amplifier chain is oscillating and that can get annoying to stop. Maybe that is what you are seeing, especially when C29/L4 is grounded compared to TP7. I would put the scope on D4-K, knowing the probe capacitance detunes things when poking around TP4,3.
Try align the AM IF into the ballpark. I disable the local oscillator and inject a weak 455kHz sine wave into the IF amplifier (mV with cap+resistor from sig gen to TP6), then adjust the tuning IF transformers for max signal. with a scope, or AGC for lowest voltage as the manual suggests.
Then I enable the local oscillator and get it ballpark, using a freq counter or luck if I can pick up a station, between the five (!) tunables - L5, and the tuning capacitor and the trims. That can be difficult.
If this doesn't work, then the radio might have an instability issue to fix or sometimes they just misbehave until aligned. The IF transformer's can needs to be soldered to ground too, at the tabs.