Hi Aussie,
I'm afraid I can't answer the question of whether your S10 will boot without a battery, but I'll see what else I can help with.
The fact that it won't hold a charge pretty much says the battery has gone to that big lithium pile in the sky and would need to be replaced if you plan on keeping the phone. Here's how you would go about that.
[guide|121826|Samsung Galaxy S10 Battery Replacement - iFixit Repair Guide|new_window=true]
The batteries themselves aren't terribly expensive; you can support iFixit's repair efforts by purchasing one from them, or you can easily find inexpensive replacements on sites like Amazon, eBay and AliExpress, among others. A quick glance through Amazon shows them selling for anywhere between $13 and $20 or so.
However, the issue here is that it's not probable that the battery is what's keeping the phone from booting. While it is possible (who knows what evil lurks in the hearts of software engineers?), there's no practical reason why your phone wouldn't boot as long as it has power from the wall plug.
At this point I'd suggest setting aside the battery issue for the time being and concentrating on the boot problem. You can, of course, eliminate the battery as a suspect by going ahead and replacing it. I'm skeptical it will help, but will freely admit I could easily be wrong.