Put more weight on something

Idiomatically, most people prefer to (metaphorically) give more weight to [some contributory factor] rather than put more weight on it. But both versions are perfectly natural, and semantically they're equivalent (attach greater importance to some factor).

Note that both "conscious agents" (people) and "abstract concepts" such as data, arguments, lines of reasoning can give or put weight on / to / behind some interpretation. But for reasons that escape me, "people" can't normally lend weight to a conclusion - only "abstract concepts" can do this.

Yes, in that context that is exactly what it means - the speaker wants to highlight the importance of the metaphysical argument and specifically wants to attach more importance to it than to other arguments. This is actually the main definition for "give weight" in the FreeDictionary:

give weight to
give weight to (something)
To consider something important or especially relevant. Often used with modifiers such as "much," "a lot of," etc.
Jonathan knew his parents never approved of his relationship with Michael, so when he decided to get married, he didn't give much weight to their objections.
Because you're a mentor here, your students are going to give a lot of weight to your opinion.

In my British English experience, "give more weight to" actually has two distinct different meanings depending on whether the subject is a piece of evidence / experimental finding, or a person.

"Finding A gives (or lends) more weight to X" means "Finding A provides more supporting evidence in favour of X hypothesis".

"Person A gives (or puts) more weight to (on) X" means "Person A considers the relative importance of X to be higher (than some unstated other thing Y)".