Should nonprofit leave bylaws lying around?

Unless you are some sort of secret society that only communicates by way of secret handshakes, I am not sure why you are so worried about members or others seeing your bylaws when visiting your facility.  Voting members have a legal right to see them.  Many nonprofits make their bylaws available in membership handbooks or in a members-only section on their websites where they can look “unsupervised” any time they want to.  Some put them on their website for all who care to see.  Bylaws are not normally considered secret.

Your Form 990 asks how you make your governing instruments (which includes your bylaws) available for others to see.  Most organizations say they are available somewhere, at least “upon request.”  That means that if you don’t let people see them, you are in violation of your commitment on the 990 and whoever signed the Form under penalty of perjury has a potential problem.  (I have no doubt that if the IRS actually pushed the issue, you would make them available to the person who had asked and nobody would go to jail for perjury, but it would undermine your credibility as an organization.)

I have no idea what you mean by violating “secure” storage.  You might want to keep an original copy of your articles of incorporation securely stored for historical purposes, particularly if they were created with a flowery flourish a hundred years ago and contain the original signatures of the founders.  But bylaws are merely a couple of pieces of paper (or now a digital record) produced in multiple copies and kept in multiple places.  You want to be sure you have a copy available for official purposes, but that is unlikely to be a single copy in a single place.

Looking at the question from the point of view of members and others, withholding bylaws from those who are interested suggests that you have something to hide?  Do you?