With a direct USB-C to 3.5mm jack input I believe you will only be able to record in mono on a smartphone. And the sound quality will depend on the (usually very poor) phone preamp.
There are no mic preamps or ADCs for recording from external devices in smartphones, unless it's an older phone with a TRRS jack which would accept a mono analog mic input on a set of wired earbuds. The mic preamps and converters for what is being discussed here will be in the external interface you connect.
For Android, I know many people have had success with this app in the past though I have never used it myself. The key points are: (1) you can use any USB class-compliant interface such as those you would connect to a PC, and (2) it bypasses any limitations or additional processing done when recording through Android the 'normal' way.
https://www.extreamsd.com/index.php/products/usb-audio-recorder-pro
I have used the same developer's playback app for many years, which also can directly stream to your external playback device.
https://www.extreamsd.com/index.php/products/usb-audio-player-pro
I can confirm the app you linked works perfectly and is well worth the six dollars. It records in stereo up to 16/48, monitors both channels independently and allows you to adjust gain on the fly. The resulting waveform looked good when I recorded music from my stereo.
For exact gear used, I plugged a pair of SP-BMC-2 mini mics into a sound pros SAPM-1-USB-C adaptor and then into a Samsung Galaxy S24 running USB Audio Recorder PRO. Next step is try at a show, maybe Grateful Drag in a few weeks but this looks like a serviceable rig that is very low profile
Finally tried this rig out tonight, quite pleased with results. I haven't listened really closely but it sounds the same as if I'd brought a dedicated recorder with me. Didn't get to try anything really loud though. Cons are that the channels are a bit unbalanced across all three bands with the right about 2.5db below the left, which is not the case when I run these mics into the standard rig. And, the app stops recording if you lock the phone screen (or some setting on my phone possibly needs to be changed). I wound up simply turning the brightness all the way down and leaving the screen unlocked all night in my pocket. Fortunately three hours of taping only used 15% of my phone battery, and that's with it giving PIP to the mics.
https://drive.google.com/file/d/1vHJ5tYOPLz9sm8NFm3iZQEBvCPIyb4Vg
https://drive.google.com/file/d/1C9O3TIEc0i0_IjT26HFBdFGuO0ZoJ_E-
gonna give it a try at a proper rock show on Saturday, will report back
I'm guessing there's a fault with that Sound Pros adapter causing your imbalance. Ask if they will send you a replacement.
As for the app stopping when the screen times out:
- Long-press on the USB Audio Recorder app icon, and select App Info
- Select App Battery Usage
- Toggle on Allow Background Usage