Why don't you just use beeminder, but set your goal to be flat? So say you're brushing your teeth, you can tell it you want to brush 0 times a week, when really you want to brush every day, and you end up with a normal-looking graph with no problem if you forget to log (because you can just backdate the data!)
Also has the advantage of letting you use it the "traditional way" (with a slope and a pledge) for a goal that can be done automatically (e.g. duolingo, word count on a writing project, whatever).
I have similar needs. I use a spreadsheet, populated via a Google Form accessible via a shortcut from my phone's main menu. I find it rewarding to make the spreadsheet display secondary metrics & graphs too.
Other popular alternatives include Habitica & habitdaily.app (iPhone only). I'm still looking for a perfect solution, but my current tools are pretty good for my needs.
Beeminder has a mode that lets you have a goal with no punishment. You have to pay for their highest-tier monthly subscription, though ($24/mo IIRC).
Good question and good answers! Someone mentioned that the fancy/expensive Beemium plan lets you cap pledges at $0. On the non-premium plan you can cap pledges at $5, so another conceivable solution is to combine that + a conservative slope on your graph + setting alarms or something? + chalking up occasional failures, if rare enough, as effectively the cost of the service.
Or like another person said, you can make the slope zero (no commitment at all), but that may defeat the point, with the graph offering no guidance on how much you'd like to be doing.
There are endless "habit tracking" apps which do this: https://play.google.com/store/search?q=habit%20tracker&c=apps
Beeminder is an excellent tool for certain things. I'm only a personal user, not associated with the company. I would encourage anyone using it to participate in their forum at http://forum.beeminder.com. So for example, if you posted there explaining your difficulties, you'd get a lot of support and ideas on how to continue to use it. As you said, you "found it extremely effective". I don't know why you were having this problem, but there have been a lot of people who have been using Beeminder for a long time, and certainly many of them have ran into the same issues you were having, and they are more than willing to help.
But yes, the answer to your question, "is there a beeminder without the punishment?" Any "to do" app or system would be similar. And there are other habit trackers, of course. Beeminder is best for things that you really want to accomplish, but are resistant to do. It's certainly not for everyone, but it's a very important and integral part of self-hacking for some people.
This might not directly answer your question, but I've read a blog post that advocated for using a similar system but if you didn't complete your goals you would instead donate the money to charity as a "punishment", which might be a solution if you're altruistic. Although you would still need to track your goals yourself; I don't think there's currently an app for this.