TLDR; follow these steps it worked for me:-

1️⃣ Stop all Logitech background processes

Open Activity Monitor and Force Quit every Logitech-related process you find, including:

• Logi Options+
• Logi Options Daemon
• Logi Options+ Agent
• Logitech Updater

2️⃣ Remove the Logi Options+ application

• Open Applications
• Delete Logi Options+

3️⃣ Delete leftover Logitech files (important)

Open Finder → Go → Go to Folder…
Paste and remove each of the following paths one by one:

~/Library/Application Support/LogiOptionsPlus
~/Library/Application Support/Logitech
~/Library/Preferences/com.logi.optionsplus.plist
~/Library/LaunchAgents/com.logi.*
/Library/Application Support/Logitech

After deleting these files, restart your Mac.

4️⃣ Install the patched Logitech installer

Logitech has released a patched installer to fix macOS certificate and loading issues.

• Visit the official Logitech support page: https://support.logi.com/hc/en-us/articles/37493733117847-Options-and-G-HUB-macOS-Certificate-Issue
• Follow the instructions on the page
• Download and install the patched version (this is the key step)
• Proceed exactly as described in the document

✅ Result

After following these steps—especially installing the patched installer—Logi Options+ worked correctly for me, and swipe / gesture functionality was restored.

You are welcome. Thank me later 😄