I agree with you generally regarding the evolution of present copyright regimes, and their lack of efficacy in their (US-jurisdiction specific) stated goal to "promote the progress...". (Note that copyright justifications in other jurisdictions differ, often strongly, see for example France with its emphasis on droit d'auteur.)

There are a few questions at play, and they may suggest a variety of different approaches. What I will note about the universal licence / tax-based / broadband-fee approach is that it seems to have been independently arrived at by numerous parties, and several of those are long-time copyright minimalists. Notably Phil Hunt, of Pirate Party UK, and Richard M. Stallman of the Free Software Foundation.

A tremendous problem in the policy field is the "getting there from here" issue, or even determining where "there" might be.

The policies which tend to be the most successful, in the long term, create entrenched constituencies in favour of those policies. Traditional copyright (and most other property-based schemas) do just this.[1]

Ideally, what you want to do is yoke personal interest to public benefit, or as I prefer to refer to it, common weal. The best of enduring welfare-state policies have done this, both in the form of direct beneficiaries and something of a support industry built up around them. (This is a case where less-than-total governmental solutions may prove more robust than conceptually cleaner and more efficient alternatives.)

In the case of shortening the scope of copyright, you might want to ask yourself what are the specific benefits manifested? and what other paths to that goal might be acceptable despite continuation of copyright?

And for the defenders of private copyrights, what are the fundamental problems which are meant to be solved?

Under the US Constitution, the justification for copyright is a way to provide income and compensation in exchange for a temporary monopoly. There are those who might say that the concept of "limited times" bears absolutely no relation to present law, myself amongst them, and I strongly suspect you. Lawrence Lessig, who argued Eldred v. Ashcroft against the 1998 CTEA's extension has said he may have made a mistake in pursuing the "limited times" clause rather than the "promote the progress" element: retroactive copyright extension has no bearing on the promotion of works already extant to which those extensions apply. Sadly, that wasn't his argument, and the extension stand.

Still: if the problem is lack of access to works further compounded by excessive costs to the public to even gain access, then a means-graded payment affording unlimited access effectively neutralises that concern.

There are other considerations: an individual might, say, wish to have a work withdrawn entirely from availablity (see, for example "revenge porn", or perhaps juvenile / early works which prove personally harmful later in life). A mechanism for total withdrawal from publication under some circumstances should probably exist. But as a means for simply denying any access to huge classes of works, and in particular abandoned or orphaned works, or those to which copyright status is entirely unclear, a universal licence would obviate those hurdles, permit compensation to a legitimate rightsholder (should one appear), and return published works, that is, works made public[2] to the public sphere.

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Notes:

1. Which suggests a ... whole vast and fascinating topic of why private-property based legal domains and polities have proven so incredibly robust over at least the past couple hundred years, if not markedly longer. I say this as no real fan of those concepts at least to an unlimited extent. I'm going to resist the siren song for now, however....

2. The etymology being, of course, "make publicly known, reveal, divulge, announce" <https://www.etymonline.com/word/publish>