Is an improvised weapon treated as similar enough to a weapon to use its properties still considered “improvised”?

In PHB, the 2nd paragraph of the description of Improvised Weapons (p. 147) states:

In many cases, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM’s option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.

If I am understanding this correctly, then the mentioned table leg (or similar object) can be treated as a club. If that is correct, then it would have the light property, as the club has it. If that is also correct, that would mean I can wield a light weapon in one hand and an improvised club-like weapon in other hand, and use Two-Weapon Fighting with those, as both are light.

If all of that is still correct, then can an improvised weapon that is similar enough to a real weapon – e.g. a table leg used as a club – still be treated as an improvised weapon for the purpose of the last bullet point in the Tavern Brawler feat (PHB, p. 170)?

Specific scenario that I am wondering about: I wield a Scimitar (a light weapon) in one hand and a table leg (treated as a club, so also light) in the other hand. On my turn, I attack with the table leg as my Attack action. Then one of two things happen:

  1. I miss – then, as both wielded weapons are light, I proceed to attack with my scimitar (per Two-Weapon Fighting)

  2. I hit – then I drop one of my weapons to free one of my hands and attempt to grapple the target.

Of course this is assuming that I have the Tavern Brawler feat, which states:

  • You are proficient with improvised weapons.
  • Your unarmed strike uses a d4 for damage.
  • When you hit a creature with an unarmed strike or an improvised weapon on your turn, you can use a bonus action to attempt to grapple the target.

Does all of that work as I’ve described, or is there somewhere a flaw in my reasoning?