I just got this working after days of trying! Using Python 3.11.
For anyone wondering, we're using a version of Python patched for iOS (from Beeware's Python Apple Support ), and PythonKit (Swift pkg) to embed Python in an iOS app using Xcode. The goal (for me at least) is to do my UI in SwiftUI, but use my own Python logic on some of the app's data.
OP, I don't know what all of your problems might be (there could be several), but I am immediately noticing at least one thing wrong. You're looking on your mac for Python when your goal is to put Python itself inside of your app. You should be looking in the app for the directory containing the Python standard library ('python-stdlib'), as well as the 'lib-dynload' subdir, provided to you by Beeware's Python Apple Support repo. You want to set PYTHONPATH and PYTHONHOME to this combo of paths.
This post is intended to supplement to "How to embed a Python interpreter in an iOS app - presented by Łukasz Langa." on YT.
In the video the OP linked to, Lukasz's 'python-stdlib' has a different name and a very different directory structure than the latest version of Py for iOS provides. What you'll need to figure out is the path to the 'python-stdlib' dir, AND [probably] its subdir, 'lib-dynload'. Counterintuitively, you do need to specify both, even though the latter is a subdir of the former.
Your line:

if let path = Bundle.main.path(forResource:"/Users/projects/extra/python_apple_support/PAS_10_11_v3/PAS_10_11_v3/Resources/",
                    ofType: nil) {

Should read something like:

if let libPath = Bundle.main.path(forResource: "python-stdlib", ofType: nil),
   let libPath2 = Bundle.main.path(forResource: "python-stdlib/lib-dynload", ofType: nil) {
        
    let mergedPaths = "\(libPath):\(libPath2)"

...where 'python-stdlib' is the PATH of the python libraries directory--not just the directory name. (If that's confusing to a newbie, this is the same: let mergedPaths = libPath + ":" + libPath2 .
To get an idea of the path, go to Product > Show Build Folder in Finder, then find Products/Debug-iphonesimulator, right click .app (the greyed / X'd out icon) > Show package contents.
Assuming you have properly copied this stuff into the project, you should be able to find 'python-stdlib' inside the .app. For me, it was right in the root of the .app (since that's where I effectively put it, by adding it to the GROUP (not dir!) called 'Resources'. (Because Resources is a group, whatever is in there will be in the root level of the .app, NOT in a dir called Resources. It seems there can be groups that are backed by folders, and groups that are not. This is an important distinction.)
Now, set PYTHONPATH and PYTHONHOME to mergedPaths like you were doing:

setenv("PYTHONHOME", mergedPaths, 1)    
setenv("PYTHONPATH", mergedPaths, 1)

If you don't include the path to 'lib-dynload', then this will only work in the simulator. Why would it work at all? Lovely question... 🤷🏽

IMPORTANT: The script.py file Lukasz created for his custom Python code must go into the 'python-stdlib' directory for this to work. I imagine you could stick it anywhere, as long as you append its new path to mergedPaths , as can be seen in a project generated by Beeware's Briefcase. To append that path to the merged paths, you're just concatenating strings with a ":" between each path. I have not tested this.
COPYING THE LIBRARY PROPERLY

Here's something that screwed me up for a while: When you copy over Python.xcframework and the std-lib, make sure you're creating folders and not groups (in the prompt after you drag and drop), else you will end up with hundreds of errors due to the flattening of directories resulting from creating groups and not dirs (think of every main.py from the lib ending up in the same folder--no bueno). Make sure your prompt looks exactly like this:
screenshot of said prompt ☝🏽

SIGNING THE PYTHON LIBRARY

Follow 'The Manual Way' > Step 6 @ Python Apple Support > USAGE.md .
All I had to change was this line segment:

"$CODESIGNING_FOLDER_PATH/Contents/Resources/python-stdlib/lib-dynload"

Due to my directory structure ('python-stdlib' in root of .app), I changed it to:

"$CODESIGNING_FOLDER_PATH/python-stdlib/lib-dynload"

...omitting the extra directories in the path to 'lib-dynload'.

ALMOST FORGOT--module.modulemap!

Assuming you have created 'module.modulemap', and copied it to all three locations (in the Xcode project and in both Headers dirs in the framework), you'll want it to read like this:

module Python {
    umbrella header "Python.h"
    export *
    link "python-stdlib"
}

...where Lukasz's read link Python , ours must read link python-stdlib , as above. I wouldn't be surprised this is actually the path to 'python-stdlib'. Again, mine is right in the root of the .app, so under that assumption, the name is also the path, here. I have not tested this theory.

NOT NECESSARILY RELEVANT TO THE SOLUTION BUT MAYBE ENTERTAINING

This part is pretty amazing. I've had this working in the simulator only for the last several days. For most of that time, on the device, it would import Random, but not Math (which was being called from Random). Then I figured how to point the app to both 'python-stdlib' and 'python-stdlib/lib-dynload'. Got a new error -- code signature invalid for '.../math.cpython-311-iphoneos.so' (that Python math module!) in 'lib-dynload'! Wow! It was finally SEEING the math module.
PARTIAL CONSOLE OUTPUT:

...(code signature in <45B34416-425D-3E01-BC39-CB7A8C170A0A> '/private/var/containers/Bundle/Application/1465B572-3399-4B76-B017-4EE168637AF5/SIXTH_try.app/python-stdlib/lib-dynload/math.cpython-311-iphoneos.so' not valid for use in process: mapped file has no cdhash, completely unsigned? Code has to be at least ad-hoc signed.)

这是令人惊奇的部分...昨天，我告诉Github告诉我Beeware的Python苹果支持的任何变化。昨天晚上7点，我收到一个通知，伙计更新了USAGE.md *，增加了代码签名的说明 *！哈！这是什么机会，会发生在我需要它发生的那一天。无论如何，我遵循those instructions，在我的手机上运行，哇！我的iPhone正在使用随机模块在点击一个按钮时生成INT！整个GUI都是在SWIFTUI中制作的。圣母玛利亚#（@）！@$它在工作！（我意识到热情和情感在SO上并不总是受欢迎的，但我会庆祝的，没有人会阻止我！）