[RPG] the price of a shield

Tower shields are not always bad. I've seen them used without being a hindrance (by someone who made a build around never having to make attack rolls). They're usually bad, though.

It's Bulky

For one thing, it weighs 45 pounds. Not a problem for a 20 STR character, but it can by itself cause encumbrance for a weaker character.

It's armor check penalty is a staggering −10. The question you mentioned had someone using it while hiding. −10 on a Hide check is nothing to sneeze at. Also applies to useful things like Tumble.

This still applies even if the shield is animated.

Attack Roll Penalty

You take a −2 to attack rolls while using it. Not a lot else needs to be said, as this just plain sucks. This applies even if the shield is animated.

Cover

You can use it to get total cover against ranged attacks. Great! Except if you do that, you:

1. Don't get the shield's bonus to AC.
2. Can't attack.
3. Don't get any protection against spellcasters, who can just target the shield instead and still hit you.

That's pretty limiting for not a lot of defense.

Few classes have Proficiency

The list of classes that get proficiency with tower shields is very small compared to standard shields. Burning a feat for the tower shield is never worth it, and one penalty for using one without proficiency is that −10 armor check penalty applied to attack rolls. You really don't want that if you ever plan on hitting anything.

Are They Always Bad?

Not necessarily. If you're casting spells that don't require an attack roll, there's significantly less downside. There are a few builds that can make use out of it in some way, but if you're a melee combatant you probably don't want that −2 to attack under any circumstance.

A general preface: each edition of D&D is its own game, and rules should be evaluated as such. Just because things happened in old rule sets does not mean that it will in 5e. Though it can be helpful to look at old editions for inspiration, you should be careful of the rules environs of the edition you are trying to modify when making house rules.

First, Shield Bash is already a named mechanic in 5E: it's the additional power associated with Shield Master that allows you to attempt to push someone as a bonus action when you make an attack. Also, you can do this same thing as a Full Action (or a piece of an attack action) using the contest rules and the "Shove" improvised action (Players Basic 74). That said, that's not what you're asking about here. What you want to know is how to do damage with your shield.

If you want to hit someone with your shield, it is treated as an improvised weapon attack.

The question then becomes does it resemble any existing weapon we have game statistics for? as per the "Improvised Weapons" section on pg 47 of Player's Basic. Looking at the list, I don't see an obvious resemblance (though feel free to make your own observations/additions). Thus it acts as a 1d4 weapon and proficiency is only granted if you have proficiency in improvised weapons.

There is also no provision for removing the defense bonus. In fact Shield Master improves the shield with attacks without removing the defense bonus so there seems no need to import that rule from a prior edition. (Additionally, it seems to be punitive beyond the spirit of 5e's rules.)

So, ultimately, shield bashing is a flavorful power that is probably not a good mechanical choice. If you want to do it, talk to your DM, but without house rules, it's not a great use of an action. If you want the mechanics, and don't mind not doing damage on the push, Shield Master's push is probably the best you're going to do.