Why would you ever use Tavern Braweler's Improvised Weapon Proficiency?

Regarding the Tavern Brawler origin feat, proficiency in Improvised Weapons means that you can pick up basically any item and treat it as a proficient 1d4 weapon.

However, this weapon attack is strictly worse than the same feat's 1d4 unarmed strike - you don't add modifier damage, you lose the free 5ft Push, and you can't choose to Grapple, Shove, etc.

Edit: You do add modifier damage, I was confused because Tavern Brawler specifies that you add your STR to Unarmed Strikes, while the rules for Improvised Weapons only specify the die size. This is poor writing and does not change my general question.

RAW, an improvised weapon can take on the statistics of another weapon which it closely resembles. However, in such a scenario, you no longer benefit from Tavern Brawler at all, since you're using your own Simple or Martial proficiencies. In fact, it's not clear to me that these count as Improvised Weapons at all. The relevant rule is as follows:

Weapon Equivalents: If an improvised weapon resembled a Simple or Martial weapkn, the DM may say that it functions as that weapon and used that weapon's rules. For examples the DM could treat a table leg as a club.

Most taverns have tables. A tavern brawler who uses items in their environment can already do this more effectively without Improvised Weapon proficiency, because simple weapon proficiency applies to clubs.

As far as I can tell, the only potential scenarios in which improvised weapon proficiency can do something which fists cannot are:

  • Ones in which nothing resembling a throwable weapon exists, and melee is impossible (somewhat situational)

  • Ones in which damage resistance is in play, and a special item with a strange damage type is available (incredibly situational - if a character is consistently preparing "improvised" weapons in advance, why not just bring real weapons?)

When building a melee character who uses improvised weapons, is it worth taking Tavern Brawler at all? Alternatively, when building a character with Tavern Brawler, why should one ever make use of the Improvised Weapon proficiency?

Side note: Right now I'm DM-ing, so what I might do is rule that improvised weapons resembling weapons never apply one's proficiency bonus, unless one has Improvised Weapon proficiency already. This gives that aspect of the feature some utility, and distinguishes it from Unarmed Strike (access to Weapon Masteries, larger damage dice). Even still, my question is about RAW. Is this feature useless?

EDIT: Yes, I'm aware that it's flavourful. Yes, I am aware that you may be separated from your weapons. However, RAW, a character without improvised weapon proficiency can already use furniture items etc as weapons, and apply their simple or martial proficiencies to those things. Doing this without the proficiency is also strictly better, since all of these Weapon Equivalents have different damage dice and require simple or martial proficiency instead.

I am not trying to powergame, if I were trying to powergame I would not be taking Tavern Brawler. I am a DM attempting to better understand the rules.

I'm also unsure why the reaction to this post's been so negative. To me, it doesn't seem substantially different to "Why would a fighter with Tactical Master choose a longsword over a rapier?", a real post on here which received a much better reception. I've seen a lot of "stop trying to optimise", "this feature is for flavour", and people castigating me for powergaming when nothing I've done implies this is the case - I have no intention of building such a character, I'm genuinely curious.