And buried in there is the seed of why I will never play a fighter in older editions...the idea that I can potentially be playing the same character for a year or longer and in that time what I can do to engage with the game is exactly the same as what I could do on the first day.
There is a lot of negative attitude towards "buttons" in this (and many) discussions, but the "buttons" are what makes this a role-playing GAME and not just a role-playing ACTIVITY.
I don't see it that way. What exactly is limiting your engagement as a Fighter in an older edition? The lack of "game-given" buttons to push? Action Surge? Second Wind? One pip better on criticals?
My Basic Fighter has zero "skills" on my sheet, and zero bonuses to anything. Yet he is still a fighter, has decent hit points for the party (5!), and engages in fights when he has to, not all willy nilly. But he will get in front of the caster (actually an elf, so a "capable" fighter on his own.)
But what buttons can I push other than attack? Hmm. I can tell the DM that I'm going to look around for tracks to follow. I can search for secret doors and traps the same as everyone else. I purposely put my "better" (in this case an 11 or 12) stat in CHA so I can get a few more hirelings/henchmen if need be, and have them be more loyal. And guess what, other "buttons" like talking to people, social stuff, keys off CHA, so my "Fighter" can act as a Ranger, a Face, whatever I want, BECAUSE I'm not tied to the buttons on my character sheet.
You only are really limited to what you can do to engage based on how you approach the game, and how your DM runs it.
I don't think I define Game the way you do. To me, a board game (start, play, end state and goals) are a game, ie. competitive. Role playing for me is not a "game" in the same way, though we are "playing" together mostly to build a story and enjoy the growth of (at least for me) my character, but not just on stats and raw power.
Heck, I'm playing a 5e Fighter and NOT pushing any of the character sheet buttons unless I really need to. Most of my activity has been (with a Noble background) going around and trying to assist villagers with various problems (bandits, cultists, etc.), and when I succeed, often with the help of those same villagers, I split the treasure with them, donate to them from my personal money, and return items stolen from them. None of that has anything to do with the "buttons" on my sheet, or being a fighter. Its trying to build my character in a Role, and build the story and background of the village and NPCs I interact with.