So in the link that you provided:
After the verb Help, you can have an infinitive form of verb. The infinitive form can be either a to-infinitive or a bare infinitive.
Bare infinitive means "just do", and to-infinitive means "to do".
[...] it's observed that in BrE the to-infinitive form dominates, though bare infinitive also occurs in BrE. On the other hand in AmE a bare infinitive is preferable.
In other words, in British English, the form you're proposing would be preferred, but in American English, it's usually the way that it currently is in game. In particular, you'll notice that "help do" has higher usage than "help to do" in a certain graph provided: http://i.stack.imgur.com/alywe.jpg
It might be a British versus American usage thing that has us differing on what we prefer. Yes, I totally agree with you on the whole that the sentence you're pointing out can sound rather strange.
Also, the "have to do" example doesn't really make any sense to mention... the usage in question is not of the word "have."
In any case, IMRy, sorry for filling your typo thread with grammar arguments...