I think how you manage your Jira is really up to you and your team. We use a different issue tracking system to Jira that has the ability to create "virtual accounts". Our last lead developer used to like all issues assigned to him which he would then dish out.
When I took over temporarily I created a virtual account called "Up For Grabs" and moved all un-assigned work to this account. I did this because I didn't want people thinking that work was being done because it was assigned to someone (me) when it was not. So I moved anything that was not being worked on to this account. Since our new team leader has joined us and we have adopted scrum, we have kept the Up For Grabs account and used that as our default assignee. We find this works best because no one in our team assigns work out, instead we pick up tasks ourselves from the current sprint. This also means no one has anything else attached to them other than what they're currently working on.
However if you have multiple teams it could potentially get confusing. I'm not very familiar with Jira but perhaps having something you can assign it to that is called something like "Team X - Up For Grabs" or something akin to a virtual user (not attached to any team member individually). This would mean your other team leaders don't get confused with unassigned work, but you also don't have a swamped account with loads of assigned work.