One argument is that a system that survives that long can't be full blown toxic (would destroy itself in the process), but it's still not guaranteed to be better than any other system that also aren't full blown toxic (include one you make up by yourself), until you dig really deep into it.
To take an example from another field, full-blown royal/aristocracies government went on for centuries in France, and the country was quite successful from several metrics. But looking back we wouldn't argue it was a pretty nice system that could still be chosen today, just because it wasn't outright toxic.
As to your point on designing your own system, I think it's just hard to design something as complex as a religious system well. Instead of reinventing the wheel, people just choose the best among the time-tested methods. For me Buddhism seems to best fit the bill. If you want examples of people designing their own Religions, look at Crowley's Thelema, or for more philosophy-focused look at Theosophy.
Everyone has a philosophical or theological code, but most people just use what they are given. Even people who don't explicitly have a system still implicitly use a value system to guide their decisions.
Buddhism was spread by missionaries throughout history. At times, much in the same way as Christianity. Undoubtedly the missionaries from both Buddhist and Christian societies thought they were spreading enlightenment and to some extent they stuck because other people agreed.
You are correct about the Buddhist missionaries, but the critical point for me is Buddhist missionaries generally did not bring weapons, whereas most Christian and Muslim areas today are so because of invasions.
A better method to compare is to look at the actual scripture. All Abrahamix faiths have some equivalent of "worship no gods before me" and various levels of actions towards this (ranging from killing non beleivers to not eating with them). The Dharmic relgions do not have this requirement, so are less likely to be used as justification to subjugate another people.
In that sense, a religion's continuity is basically that group's continuity. Individuals within that group can switch to another religion, same way people move countries, but I'm not sure there are any case of a group switching wholesale to another religion by it's own decision. A king changing faith could be the closest to that, but I think it's usually a political move to join a bigger group, more than just independently switching for the sake of it.
> designing your own system
In general I don't think a "system" needs to be complicated, all ecompassing, nor needs to be fully from scratch. I'd compare it to cooking, you could look at a steak and carrot recipe and decide to try arrangements, and end up with mushrooms instead of the steak. You're still not building a new school of cooking, just doing your stuff on your own, eventually mixing and matching recipes you've seen elsewhere, without having to tag yourself with any specific cooking style.
"The vast majority of U.S. adults (85.6%) say they were raised as Christians. But more than a fifth of them (19.2% of all adults) no longer identify with Christianity." [0]
I think a ~ 20% change within a single generation is pretty significant. And most of these people will use the 'recipe' method you describe because it is easier than learning an entirely new system. There are many historical examples of this. E.g. Taoism + Buddhism -> Chan Buddhism -> Zen Buddhism.
[0] https://www.pewresearch.org/religion/2015/05/12/chapter-2-re...