In 1969 Avalon Hill must have been seeking to up their game. Their 1960's designs were stuck in a rut with the likes of Stalingrad, Battle of the Bulge, Guadalcanal, Gettysburg, D-Day, and Waterloo. So they took a chance on some new blood with free-lance designs such as Anzio (Dave Williams) and 1914 (Jim Dunnigan).
The first edition of Anzio was indeed confusing. For starters, the counter set had separate (single-sided) game pieces for each step level of every combat unit. This was an emerging wargame mechanic in 1969, quite new to the Avalon Hill faithful. The basic game of the first edition used a traditional CRT (D Elim, A Elim, Exchange) and only the full-strength counters. So the great majority of counters were not used in the basic game as first published. It was not until you got to the advanced game that you switched to the step reduction CRT and the organization sheets became necessary to track all those game pieces.
The original rules for the Italian order of battle were a mess, considering how little impact they had on the campaign. The mapboard had way too many ambiguities that were not addressed in the rules (as they were in later editions).
I'm about the same age as the reviewer. I owned a copy at the time (I was 16) and remember the confusion and the many requests by Anzio owners for clarification from Avalon Hill. Back then AH referred to rules questions as nutmail. You had to include a stamped, self-addressed envelope to get a reply, and you were requested to ask only yes or no questions.
Errata was eventually published - possibly arriving with my next copy of The General. I remember literally cutting and pasting errata into the rules. After incorporating errata and clarifications, I played Anzio several times (opposed) and remember being completely absorbed in the game-play.
Another devoted fan of the game, Tom Oleson, volunteered to take over development of the game for Avalon Hill. He re-wrote the rules and charts, making the game much easier to get into (in my opinion). Sales did not improve, so the last version published by Avalon Hill was the 4th edition (1980).
Sorry...but I can never look at this box cover without thinking about the Dr. Seuss movie: The 5,000 Fingers of Dr. T
Tom Oleson's development of Anzio continued into the early 2000s and the game was renamed Italia, to distinguish it from his add-on scenario, Sicilia.
Now about the video review...
The Anzio box shown in the video is the 4th edition, so the rules should also state "4th edition" in a box on page 2. If not, then Gil's got a mismatched rule book with that box. (The counters stayed the same from the 1st to 4th editions.)
Early on, Stalingrad is mentioned with fondness as a WWII game that was fun to play. If there was any expectation that Anzio would be Stalingrad in Italy this could understandably lead to disappointment, since Anzio is more sophisticated than Stalingrad or any other WWII game published by Avalon Hill at the time. It was the first game by Avalon Hill to have multiple step reduction counters per division, for example. (Guadalcanal  had a step reduction roster sheet.)
The reviewer understands that it is a step-reduction combat system, but seems to complain because the backs of the counters are blank, and there are too many counters. Or perhaps not enough counters?
The Basic Organization Charts depict every counter used in the basic game. This is noted in basic game rules C.1.C and C.2.C:
Space is provided to set up all the counters needed in the Basic Game.
Of course you will have counters left over, since they're not all used in every scenario. The upshot of the entire video seems to be "I can't set up the game" after many hours of trying.
As for not knowing whether it is even a complete game - there are scans of the the counter sheets on BGG so an inventory would be simple enough. Hard to see it would require hours to verify. Who buys a used game without checking for completeness on arrival?
Here's what I'd advise:
- Place matching counters on every space of the Basic Game Organization Chart. Ignore the Order of Battle area of the Basic game chart until the next step. Set aside any leftover counters - they are not used in the Basic game.
- Move counters from Organization Chart to Order of Battle, matching unit designations and combat factors.
- Move Order of Battle counters to the map as indicated - specific hex or a named location. Counters labeled with an arrival turn stay on the Order of Battle until the turn specified.
- When taking a step loss in combat, the unit taking a loss is moved to the Organization Chart and its next lower step counter is placed on the same map hex. The Order of Battle is never involved in the step reduction procedure.
If you have trouble with the basic game of Anzio, I have my doubts that you're ready for the Simonitch "4X" system.