Traditional highly intense low-pay professional jobs (university teacher, social worker, clergy, for example) have, at least in some cases, sabbaticals as part of the career and compensation scheme. Sabbaticals have the express purpose of letting people avoid and recover from burnout.
Highly intense high-pay professional jobs (oncologist, software developer, bond trader) don't have the sabbaticals baked in. Those of us with that kind of job have to cope with burnout on our own. There's no well-meaning dean or bishop saying, "go take a sabbatical. I want to see a five-paragraph paper describing the insides of your eyelids. It's due in three months. Getouddahere! Go! Now!"
Nobody's going to do it for us. We have to do it for ourselves.
You say you don't want to let your team down. That's great. Keep in mind that they don't want to let you down either. So trust your team members to keep doing what they do, but without your presence for a few weeks. Talk to some of them about it.
Then talk to your managment / front office about it. Come to that conversation with a tentative plan. Say something like this.
"I am fried. I propose to take six weeks off and go snorkeling in the tropics / cut my grass with scissors / build a bicycle of bamboo and ride it to Taos NM / whatever.
"Joey and Anne know what needs to be done in the farkle and framis modules, and they're well-respected by the rest of the team. I know this team; I'm confident they'll continue to do great things while I take the break I need to take.
"What do you think of this plan, boss? Is it workable? Would any changes make it workable for you?"
That opens the conversation in a respectful way. Good managers will probably see how the company benefits ... give responsibility to more people ... keep you "in the socket" even if you're powered down for a while ... build a reputation as an employer with respect and compassion.
If your proposal angers your boss or starts a riot in the executive washroom, well, then you know that about your company as you consider your next options.
A blunter way of putting this is "Always work yourself out of any job you hold. It's the only way you'll be able to do anything new."