HOA President a renter

We can't see your CC&Rs, so nobody can give you a definite answer, but:
1) Most HOA's set ownership as the primary metric for membership in the HOA
2) It is the norm to require that Board members be members in good standing in your HOA (declerant control not withstanding), and voted into place by the general body
3) Officers are chosen from the Board.

Hence, your renter fails test (1), they cannot pass onto (2) and (3).

So the answer to your question is likely a resounding NO

My HOA governing docs are a good example of this, they don't require directors to be owners/members. In this case it's up to the members to decide at election time if they want non-owners on the board. Once on the board, it is up to the directors as a whole to elect the officers, so in theory it could happen. In reality, I doubt it would because most owners probably wouldn't vote for a renter, either way it's democracy at work.

Why oh why would anyone want to have someone not owning property in the community serving on a board that manages this same property?

They would have no financial interest in the property but none the less be placed in a position to make decisions affecting the property.

Whether allowed under the documents or not I would never support a none owner holding a board position.

Not sure how the process works elsewhere but here in order to be placed on any ballot the person would have to appear on our owners listing.
In 29 years I have known the status of any board candidate. Simply check records.

If not an owner they would be ineligible.

Again, no matter how many open positions a non-owner would not be permitted to fill any vacancy. In the case were the number of candidates was less than the number of open positions we would be left with a board vacancy. We would not then allow someone not owning property fill that vacancy.

We simply don't allow important matters like board membership to " fall through the cracks" because no one bothered to check.

"Not sure how the process works elsewhere but here "

My guess anywhere the owners would have the ability to keep non-owners from serving. Would require they involve themselves in the process and concern themselves with who is overseeing the management of their investments.

Feel need to mention this as it happened to us. When we were re-drafting changes to our documents, one change was to remove all references to the developer. This meant we removed any developer addresses, 2 voting system, and who to send the check to. However, due to lack of fine tooth combing it was missed about a reference to being an owner only being members thus serving on the board.

This was only one real sentence in a paragraph. However, it really was in reference to the original developer being in charge. It did not apply to when the owners were the owners of the HOA. This caveat was for when the developer put in their own board of their choosing. It allowed them to hire or use their own employees or Management company. Think about it. A developer, their employees, nor the MC are members of the HOA. (Unless developer owns a home). If this reference wasn't in the documents, then they could not put in a preliminary board.

Now, when the HOA is turned over to the owners, it is often not updated to reflect all the necessary changes when developer controlled. Our documents referenced the original builder for over 20 years because no one ever thought of updating/redrafting our documents. It was when we changed water meter to separate ones it had become apparent we needed to remove Developer references.

So if your documents haven't been updated to reflect no longer developer controlled or not all references were removed, then it may lead one to think one does not need to be an owner to be a member of the HOA. I would run this one by a lawyer to make sure it's really not an overlooked change/reference. It's best to be able to define what makes a member and/or can run for the board.