Below is Adams picture with voltage readings in TX mode on my kit. We already established it works on receive.
Now, I am assuming that your kit goes into TX, the relay clicks and the TX light comes on? You can keep it in TX using the APP, don't forget to set Boost in the settings. Use your cable to connect your phone to the audio in jack, use a dummy load and squirt 1khz into the rig.
The rig should go into TX mode and stay on so you can do your checks.
Test Q8 first, this enables the xtal to resonate in TX mode, it basically shorts to ground to give your crystal a ground.
If this checks out we can assume your rig is oscillating at your crystal frequency, you can check also with a short wave receiver.
Next move on to the buffer amp Q2, this buffers the pre driver from the mixer and presents an impedance the mixer is happy with, you should be getting 5v on the collector which is fed from the little regulator on the board. If you have a scope you can follow this signal path through and hopefully watch it grow.
Now move to Q3, this is the start of the business end and boosts your AM signal to a level capable of driving the PA final transistor. Again the collector should be near the value of your power supply after dropping a little via R7.
Next you can check Q4 which again should be at your supply voltage on the collector. Check you mounted it so you can see the writing on the front. Check you have continuity through L1 if no volts present, sometimes you need to scrape the enamel off the wire pretty hard.
If all that checks out we have to look at T1, its a simple coil but 3 ways to get wrong, again check the enamel is cleaned off properly for a nice clean joint.
After all the above and assuming all your resistors are in the correct locations we have no choice but to think something is smoked. The Final transistor is the most likely target since for a strange reason when you power up the rig the TX comes on for a second. This can catch you out if you haven't got a proper load connected. PUFF.
By far the best way is to clip it out, this way you are only unsoldering one leg clipping at a time and preserving the traces on the board. From here you can test the transistor with a multimeter or tester to determine if it was smoked or not and replace with another after checking that one before fitting.
I am confident you can get it chooching at least to save some sanity. The power out will be low, you may struggle even with a QRP meter, a scope is your best tool for measuring here. If its all chooching and you can see yourself on the return beacon you can then plan how you are going to increase the output.
Make a note of all your readings which will help determine how that can be achieved. They may not match mine due to tolerances and your supply voltage but they shouldn't be far out.