Again, there are numerous definitions of "right". Most usages in this thread seem to fall under the OED's 9th definition, of "a legal, equitable, or moral title or claim to the possession of property or authority, the enjoyment of privileges or immunities, etc."
Whist legal rights might have some enforcement mechanism, equitable or moral rights (divorced of legal aspects) typically would not. The right is recognised or might be asserted or defended, but by other-than-legal means.
Black's Law Dictionary gives a number of definitions, though as these are (largely) specifically in the context of law, their narrowness is somewhat expected. Fundamental right however has as its first definition "a right derived from natural or fundamental law", which might be construed as at least partially exceeding legal enforcement.
The whole notion of rights can become complicated, and whilst I often agree with the sentiments or goals of those advocating for certain rights, I find the specific rationale, logic, and/or empirical grounds often weak, leaning far more on rhetoric than some basis in reality. At the extreme, for any given right, based on "natural" or "fundamental" law, it's virtually always possible to construct a competing right which negates or countermands that.
The rights of speech vs. privacy, of bearing arms vs. freedom from coercion or fear, of access to healthcare vs. freedom from supporting another, of the national right to defence vs. the right to refrain from violence (including supporting it monetarily through taxes), etc.
There's a school of thought which dismisses the notion of rights, probably most famously Jeremy Bentham. I'm not sure I fully subscribe to his views (I've only read brief summaries, and don't substantially know them), though I'm inclined that way myself.
What I see are competing sets of freedoms, privileges, responsibilities, and obligations, most of which exist, as you suggest, based on the ability to assert or defend them as a practical matter, and to that degree I think we are in some agreement. I'd be more willing generally to suggests rights in a moral sense that should be aspired to. These might be your unenforceable, but not meaningless, rights.
There's a tremendous amount of historical reletivism and present-bias in discussion of rights. There've been incredibly durable and arguably thriving societies whose rights and values systems differ sharply with those of most present-day countries. There's been a considerable movement in questions of ethics, morality, and rights within my own lifetime, within my own homeland, and those developments are far less than those experienced elsewhere over the same period.
Absolutist declarations of rights tend to end poorly.