I don't think anyone can truly answer this for you except yourself. Your management believes in you. You have to decide whether it is worth the stress that goes along with it. I can say that with an accepting environment, a stutterer can succeed in a leadership position; I have seen it, and in positions which required public speaking on a regular basis!

Why not tell us what is on the other side of the coin? What do management see in you that makes them offer you this post?

"A stutter over a strutter." Most people would very much deal with a stuttering manager than a pompous and loud one. If you're a nice, kind and professional person, your employees will have your back in no time, stuttering or not.

Did you know that the current POTUS has a stutter? He did not have to decline "promotions".

I don't know how this would come across, but maybe ask to be manager for a limited project only at first? Like Project X has a point at which it is completely and clearly finished in like half a year or a year. Maybe you can just take over that and see how it goes regarding your managing skills

Sounds like you're doing an awesome job already and that no-one seems to care about your stutter. Go for it! Everyone has their quirks, and generally in a team those quirks become invisible once they are no longer "new". Enjoy your new role!

Agreed with @Basya - this isn't a question, you've just told us a story. We can't make a career decision for you and you have the best information to make this decision...so it's going to be up to you to make that decision. Making decisions is what management is all about. Maybe if you can't make this one, you're right to doubt yourself. If this one decision is giving you sleepless nights, get ready for lots more of them when decisions fall on you day after day.

This site is very strict about questions having the correct structure, and is likely to eventually close this one (even with highly-voted answers) unless it can be reworded to ask a specific question (e.g. "What are some effective strategies to overcome a stutter when working as a manager?"). This would also help ensure you get the best answers. As others mentioned, you have the best insight into the pros and cons of a career change; but some people here may be able to give you good tips on dealing with your stutter in the situation in question.

That info could help you make the best decision for yourself. I hope my two comments are helpful. This looks like a very important question to you, and I'd hate to see it get closed--I think it can be turned into a good (from an SO standpoint) question (one that benefits you and others in your situation), and that your doing so would make it more beneficial for you as well by giving you focused and concrete information that can help you decide if this is something you want to try.

My worry about this being closed isn't theoretical FWIW--there are already 3 close votes as of the writing of this comment, likely (just a guess) for the reasons I've mentioned in my comments.

My manager said not to worry about it and if anyone has problems just advise him and HR - This is not about "tattling" or "running to the boss". This kind of harassment is dangerous for companies and it can't be tolerated. It must be reported if the company is to be allowed to protect itself. While you'll be able to handle the harassment in your own right, if the company doesn't know it's going on they won't be able to act on it effectively.

What could possibly be the fallback of this? If this happens to actually be an issue for being a manager in this company, at worse you can probably be back where you are now, with lessons learned on what you could improve (and it may not be stutter-related at all) to fulfill that role in the future and/or somewhere else

I didn't see any actual question in this post. So I added one. OP - please change it if that doesn't capture your intent.

"Everybody stutters one way or another, so checkout my message to you--as a matter of fact, don't let nothing hold you back, if the Scatman can do it, so can you." -Scatman John

You say you like your current position, and you don't seem particularly keen on switching to management except for the "what if" factor.