Why wont my mac allow a larger partition?

I think the best you can accomplish is to first, select the small partition. Since it won't let you delete that partition, try grabbing the left sizing gadget and dragging it toward the topmost gadget until the partition is as small as you can make it, then click on the Apply button. You could also try selecting the larger partition, then change its size to 500 GB. Click on the Apply button. One of these should get your drive back to its full capacity.

Understand, that you do not need to create a small partition on which to install the macOS system files. They are already installed on a small volume in the main container. This small volume, however, shares the space in the container with all other volumes. So, it will show the same capacity as the other volume. This is normal in the APFS filesystem of Catalina.

Add, delete or erase APFS volumes in Disk Utility on Mac - Apple Disk Utility User Guide

When should you use APFS Containers, Volumes, and Folders?

Disk Utility can’t resize APFS disk images, but hdiutil can

I think the topmost picture is how the disk is really configured. I would leave it alone. I can't answer your second question without asking what you would do with this new partition. Fill me in on what you would use the space for, then I may be able to come up with a workable solution.

Your drive is not formatted properly for Catalina. Catalina configures two main volumes: Macintosh HD and Macintosh HD - Data. In addition, there are three other small volumes: Preboot, Recovery, and VM. Only Macintosh HD is visible. Is your internal drive an SSD or HDD?

You cannot enlarge the small partition until you shrink the larger one. At present, it does not appear that Catalina is actually installed on the drive. My suggestion at this point is to delete the small partition which should restore the drive to its full capacity. What version of macOS is installed on the drive, presently? Is it really Catalina? If it is, then does Disk Utility give you this view of the drive?

In the upper left corner click on the View gadget and select, "Show all devices." You should see something like the above but for your HDD. Is that what you see?