Here's the deal:
If I type "Psalms" in the headword field in LBD, I get a list of suggestions that match right away.
In OTP, however, I don't. "Psalms of Solomon" is only suggested once I have typed "Psalms of Sol".
The OTP is showing Milestones, not Headwords. I know that doesn't help you, but it at least explains why it's different. In the meantime you can type PS Sol to save a few keystrokes.
The OTP is showing Milestones, not Headwords
Indeed. Thanks for the correction.
Are Milestone suggestions supposed to be "slower" than that for Headwords?
No, but they work differently.
Headwords are simple, in that if you start typing P, it's obvious that you want a word beginning with the letter p. More over, you'll want to start reading at the beginning of the article, so selecting the suggestion is useful, because it takes you straight there.
Milestones are much more complex. Even in the Bible Milestone, if I type a '1', I could mean a bible book like 1 Chronicles, or I could mean chapter 1 (or 10-19, or 100+) of the current book, or verse 1 (or 10-19, or 100+) of the current chapter. If the book has multiple milestone indexes (as OTP) does, there are even more options.
As a result, the normal behaviour with Milestones is not to make any suggestions enough characters have been typed to disambiguate the query (it's one character with headwords). So, in a Bible, you only need two letters to show the suggestion for Ecclesiastes or Exodus, but you need three characters to show the suggestion for Mark (it doesn't suggest Mark, Matthew or Malachi after two characters).
Moreover, most of the time when you're typing in Mark or Ecclesiastes, you don't want to go to the beginning of the book, but to a particular chapter. Therefore having a suggestion that takes you to the beginning isn't necessarily as useful anyway.
It would be nice if it worked like Google suggestions. In the latter, I can type "Gospel of" and receive the choice "Gospel of Mark" and then instead of entering to search for "Gospel of Mark," I can use the down arrow to navigate to the suggestion I want, supplement it with what I want, then press enter. Similar behaviour would allow us to see Malachi, Mark, and Matthew when we type "Ma," quickly select what we want and complete it with a reference if needed. Of course, this is only an example, we don't really need this for Bible navigation, but for other works like OTP, it would be useful, I think.