You CAN totally run your Mac without the CR2032 battery!
(On a Mac Pro 3,1)
I was in the same boat as everybody, and the broken clip.
With overall exceptional design, I was really surprised that they under engineered the clip. On first view, it looks like it should just fold up & off, so I was really surprised how easily it snapped off.
Before taking off everything to attempt a complicated soldering repair, I thought, why not just power it on & cross my fingers?
Happy to say that it works just fine, and infact it fixed the issue I was having where the system had stopped detecting my wired keyboard/mouse & anything USB.
That was my initial reason to replace the battery.
In essence, YES you will be fine running your system without the battery.
It actually booted faster than ever before, which is an added unexpected benefit :)
& best of all...after numerous other attempts, my USB ports and keyboard finally work!!
By all means, leave the battery off.