Welcome back to the QuickBooks Community, LaterWhen.
I appreciate the explanation of the features you're using in QuickBooks Online to enter your supplier transactions and definitely see how being able to record the payment method can make a difference in how you're tracking your books. I can go over this with you so you can have a clear sense of expenses versus bills and know how you can move forward with this.
While you're correct that both expenses and bills are somewhat the same in that they're transactions to your suppliers, there are some key differences between them. Just like a sales receipt and an invoice, the difference between an expense and a bill is an immediate payment with the exchange of goods/services versus a more delayed payment. This article goes into more detail if you'd like to read more: What is the difference between bills, cheques, and expenses?
Regardless of which you need to go with, you're correct in your observation that the Expenses form includes the Payment method drop-down to make a selection while the Pay bills window does not. This is how the program is designed and QuickBooks Online registers bill payments as cheque or credit card transactions by default based on the Payment account chosen. For instance, if you choose a credit card account, the transaction will show in your supplier transaction list as Bill Payment (Credit Card).
There isn't a way to change this otherwise at present. What you can do is use the Memo field if you'd like to note the bills were paid in some other way. That field can be added to many reports through customization, which can still help you see these details.
There are two additional recommendations I have for you. The first is to submit feedback about changes you'd like to see to the Pay bills window by taking these steps: How do I submit feedback? This helps our product development team learn the ways that users would like to be able to use and log their information in QuickBooks Online. To stay informed with new features, check out this page: QuickBooks Online Feature and Product Updates
In the meantime, you can consider using a third-party app to track your bills more precisely. The Apps tab in QuickBooks Online allows you to search for partner applications that can perform various features. I suggest visiting that tab and then using keywords like "bills" to see if there's one that offers a fuller bill management system for your needs. Each listing contains information and contact details for the app developer, and many also have reviews from other users.
I hope this has been helpful! Take care and have a great week. :)