Hello Keith Nelson,
Hope you are doing well. Thank you for reaching out to the Microsoft community.
I read your thread and get some basic information of your scenario. To better understand and investigate your issue, I may need some more detailed information from your side so that I can narrow down the issue in right way:
May I please know if you are using Microsoft 365 business/education email account, Microsoft personal account such as Outlook.com, hotmail.com or live.com, or other third-party/custom domain email account?
How many accounts added into Outlook, and which one is the default account? (If you have multiple accounts and set a personal email account as the default one, would you please change the default account to your business one, and check the issue again.)
Will you please share me your Outlook version details? Please follow this article What version of Outlook do I have? - Microsoft Support and follow the steps to check.
To view your task list on the same page as your email in the new Outlook, you can follow these steps:
1. Sign in to "New Outlook".
2. Select the "Tasks" icon at the bottom of the page.

3. In the Tasks pane, you can see and manage your tasks. This is where you can create new tasks, view task lists, and manage task details.

4. To view your task list on the main email screen, you can use the "Pin" feature. Click on the pin icon next to the "Tasks" tab to pin it to the main screen.
5. Once pinned, you will see your task list displayed alongside your email in the main screen.
Please understand that our initial response may not always resolve the issue immediately. However, with your help and more detailed information, we can work together to find a solution.
If above mentioned steps do not work, kindly share some screenshots on clicking the app icon. Then we can do further testing and investigation accordingly.
Thank you in advance for your understanding. Your patience and cooperation will be greatly appreciated. I hope for all the best!
Sincerely
Nafis Iqbal Shuvo