Here's how we intended the system to work at one of those big ones-
We were using the sky blue devops system that the company developed to track development tickets.
Bugs were triaged as soon as they arrived, by email or otherwise, and they were either fixed asap or turned to a development ticket with the needed priority.
If you think about it there is no real reason to track bugs as such, hopefully your management won't care about bug statistics.
The above system works well enough if it's a small enough product and not too many bugs, if you have to track bugs anyway you can use your development tickets tagged with "bug"
We use jira and bitbucket for that. I'm happy with it.
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Have you looked at Trello? I think it's good for small, rapid-fire issues. Each card on a Trello board can be "flipped over" to add substantially more detail, too - so it scales nicely to larger tasks that need to traverse the same pipeline.
My team uses Jira, but I'd consider migrating to Trello if our workload looked like what you're describing.