How to Link/Embed an Objective-C Static Library of Extensions to an Xcode 13 project?

Frankly from the steps you provided I can't spot any specific issue. The only part I would do different is the universal binary, since I just can't see a reason to do that (yes, it will require you to re-compile the library when you switch to another platform, but only once. Also you will have "less fat" final binary and freedom to change the source code of it on the fly). Let me just sum up all steps with instructions below:

Sample Project

Our sample project is the iOS project template built by Xcode 13 for iOS applications in Objective-C with the following file structure:

Static Library Project

Provided the IDE used is Xcode, the steps to create a project as part of another project are as follows:

1. While having the main project opened, open the File / New / Project menu
2. Choose the iOS / Static Library template and go the next step
3. Give the project a name (I gave SLProject) and go to the next step
4. On this step choose the root folder of your main project and ensure you have the main project specified as the target for the Add To: option

You should end up with the project structure similar to this:

It's not necessary but for static libs I'm used to make one "master header" which has all other library interfaces included (similar to the umbrella headers for frameworks) with the name of the said lib (SLProject.h in this case), so I remove the implementation file for it (SLProject.m) as well as content of the header itself for now (there is nothing to include). Now, let's create a little category for UIView which prints out the view itself and all other views in the hierarchy (if available):

UIView+PrintHierarchy.h

@import UIKit;

NS_ASSUME_NONNULL_BEGIN

@interface UIView (PrintHierarchy)

- (void)tdw_printHierarchy;

@end

NS_ASSUME_NONNULL_END

UIView+PrintHierarchy.m

#import "UIView+PrintHierarchy.h"

@implementation UIView (PrintHierarchy)

- (void)tdw_printHierarchy {
    NSLog(@"%@", self);
    if (self.superview) {
        [self.superview tdw_printHierarchy];
    }
}

@end

Next, I include this feature in my "master header":

SLProject.h

#import "UIView+PrintHierarchy.h"

And ensure that all headers are included in the Copy Files phase of the static library target:

Linking the projects

Now go to the main project target's Build Phases.

1. First, you need to link the main project with the static library target under the Link Binary With Libraries phase (click + and find the target of the static library project we just created).
2. I also usually add the projects my current project depends on in the Dependencies section
3. Since we deal with category methods compiled as part of a static library, due to a known issue, don't forget to add -ObjC flag to the main project's target linker options under Build Settings

For Objective-C part that is it! From here you just import the static library headers wherever you need to and use the methods defined there:

#import <SLProject/SLProject.h>
#import "ViewController.h"

@interface ViewController ()

@end

@implementation ViewController

- (void)viewDidLoad {
    [super viewDidLoad];
    [self.view tdw_printHierarchy];
}


@end

Linking Swift classes

In order to provide access to the same library for Swift classes, it's enough to just add this "master header" to the bridging header in your project:

iOSStaticLibDemo-Bridging-Header.h

#import <SLProject/SLProject.h>

And now everything is just made available to the Swift classes as well without any extra imports:

import UIKit

class SwiftView: UIView {
    
    func someMethod() {
        tdw_printHierarchy();
    }
}

Feel free to use the project created while I was writing these instructions here as a reference.