Composting fail?

I think I may have screwed up my first attempt at composting. I started with a tiny pile of matter not even in a container, just in a random spot that gets sun all day. Gradually I added to this pile. Cut up veg scraps, crushed eggshells, coffee grounds, tea leaves, weeds, paper etc, and I watered the pile daily. But even with heavy watering, the Australian sun was drying out the matter very quickly. I recently transferred the pile into a container with holes, but no lid. Despite all the watering and the fact so much of the matter was fresh or green when it went in, the pile now just looks like a heap of dry twigs. To put into perspective how strong the sun here is, I recently weeded and put all the weeds into a bucket. Just a day or two later in the sun, the weeds were crispy.

I'm guessing because the moisture was never locked in, this is why the pile turned out this way? If I start over, can I use this same pile as part of the brown matter? Or has everything beneficial been killed in the sun?

Move your compost to the shade, and also compost piles won't get hot unless they're big, like cubic meter big. Otherwise it sounds like what you're doing is right. Yes you can use the same pile to start the new one.

As stated get some shade over it or cover it.  Keep an eye on those temps and turn the pile as needed.

Here is way to much information on composting
Composting compilation

It talks on compost, compost extract and compost tea.

I also will say to consider vermicomposting as I believe generically from what you written was in part issue watering it in.  If you do not want seeds in vermicomposting you simply start the feed stock in a compost pile (what should be done with manure) until it gets to proper temps for 3 days or so (I speak generically on the time) and then you stop the composting process and spread the compost out and let cool and use that feedstock in your worm bins.

I look forward to seeing success in your future composting workings!