A successful forum launch is, in my opinion, a working, publicly available forum. That's the launch.
These days, I can't see a forum growing from scratch without some kind of boost.
I was lucky because I launched AVForums while I was managing some editorial sites (in 2000) and I sent traffic to the forum from those sites and also back to them.
In 2019 I launched a forum for MG EVs and the boost I got there was a link or two from the existing Facebook group.
I could also have sent MG dealers leaflets for them to distribute with all car test drives and sales in exchange for some exposure on the site. But I didn't need to.
It's definitely worth thinking hard about how to get traffic to your site and maybe investing financially in that.
One metric for success you could use might be that the income from the forum sustains it.
Whether or not that includes your time depends on your approach.
These days it's probably hard for smaller forums to generate much revenue from traditional banner ads after the tracking cookies are being phased out and the value of programmatic advertising has dropped through the floor.
Another metric might be an arbitrary number of members. Maybe there is a competing forum and your goal is to get more members than them.
I think each admin will have to consider what success means to them.