Why doesn't JIRA have a Severity field like Bugzilla?
Originally, JIRA did have both a Priority and a Severity field. The Severity field was removed for a number of reasons, but principally because it was confusing to business users. To a software developer, it seems obvious that the severity of the bug ("The system crashes completely") is unrelated to the priority of it ("There is a one in a million chance of this occurring"). However, JIRA succeeds so well because business users can actually use it. If you present a business user with these two fields, they are instantly confusing (which is why the Severity field was removed).
In order to re-implement Severity, you can create a select-list custom field, order it (with field layouts), put it on your filters (with column layouts) and indeed search and filter it (in the Navigator).
For details, please see Custom Fields.