What is the best use of a Shield and Longsword combo?

I received this inquiry the other day: “I like to play a Paladin that often uses a Shield and Long Sword combo. However, I have noticed that there is no shield bash in 5E. I have been using the shove instead as an action between a Trip and Shield bash. Are you aware of any attack that tries to leverage a Shield in 5E? I would really like to stress a shield proficiency for my build if possible.”

Here are my thoughts:

A shield is an improvised weapon dealing 1d4 bludgeoning damage.

Paladins are proficient with shields. Whether said proficiency extends to it being used as an improvised weapon is not specified, but there’s no reason why it shouldn’t be.

Note that shields aren’t light weapons, and thus can’t be used for normal two-weapon fighting.

If your Paladin adopts the “protection” fighting style at 2nd level:

“When a creature you can see attacks a target other than you that is within 5 feet of you, you can use your reaction to impose disadvantage on the attack roll. You must be wielding a shield.”

If your group is using feats (I think most are) there is the “Shield Master” feat:

You use shields not just for protection but also for offense. You gain the following benefits while you are wielding a shield:

If you take the Attack action on your turn, you can use a bonus action to try to shove a creature within 5 feet of you with your shield. (If I was DMing the game, I would allow the use of this bonus action to either shove a creature or to attack it with the shield as an improvised weapon.)

If you aren’t incapacitated, you can add your shield’s AC bonus to any Dexterity saving throw you make against a spell or other harmful effect that targets only you.

If you are subjected to an effect that allows you to make a Dexterity saving throw to take only half damage, you can use your reaction to take no damage if you succeed on the saving throw, interposing your shield between yourself and the source of the effect.

And the “War Master” feat will allow you to cast spells while holding both your shield and sword.

I would like to hear if anyone has any other thoughts on this matter.