No.
You can't take a bonus action during another action, thus, you cannot take the shield push's push bonus action until after you've used your attack action.
To address the possibility of taking it before you take the attack action, the power seems to specify that it happens because you take the attack action. If somehow, the shield push invalidates your ability to take the attack action (like, perhaps, you push the only available enemy off a cliff with the push), you've invalidated the ability to take the shield push action in the first place.
No, you can't use the bonus-action shove to attack in any other way.
As you noted: Per Jeremy Crawford's clarification on Twitter and subsequent 2019 update to the Sage Advice Compendium, the bonus-action shove from Shield Master can only be done after completing the entire Attack action.
The first bullet of the Shield Master feat (PHB, p. 170) says:
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.
The word "shove" refers to a specific thing in 5e, described here in the rules:
Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you. If you're able to make multiple attacks with the Attack action, this attack replaces one of them.
The target must be no more than one size larger than you and must be within your reach. Instead of making an attack roll, you make a Strength (Athletics) check contested by the target's Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). You succeed automatically if the target is incapacitated. If you succeed, you either knock the target prone or push it 5 feet away from you.
The feat makes an exception regarding the timing of your shove, but everything else about it is the same. The mechanics for that shove are collectively described in the feat and in this section of the rules.
The rules on improvised weapons describe the mechanics of attacking with something that is not a weapon or otherwise designed to attack with. They don't describe the timing of such attacks.
The Attack action lets you make an attack (or, with the Extra Attack class feature, multiple attacks) using your action. The rules don't specify what you have to make this attack with, so you can make these attacks with any combination of weapons (normal or improvised) or unarmed strikes.
No rule allows you to substitute that regular bonus-action shove (granted by Shield Master) for any other sort of attack or action - including the use of your shield as an improvised weapon to attack (and do damage). In the absence of such a rule, you can't do so by RAW.
(As always, your DM could house-rule otherwise, but this would essentially be granting you a free attack every turn.)
Best Answer
The action cost is different, but the effect is the same
Shield Master lets you shove without sacrificing an attack
Normally, when a character takes the Attack action, they can replace any of the attacks they get with a Shove action. Shoving in this way involves the sacrifice of one attack.
With Shield Master, the character need not sacrifice any of their attacks when they take an Attack action. Instead, they can shove using a bonus action.
They are very different things. A character who gets only one attack as part of an Attack action must give up all their attacks to shove someone. That is a significant sacrifice. Getting the same action as a bonus action means that one need not make that sacrifice. Which is a distinct advantage if that is something that the character is doing on a regular basis.
Effects of the shove are the same
Regardless of the actions used to take it, the character is still taking the Shove action. As such, the effects of the Shove are exactly the same.
Also, note that having Shield Master does not prevent you from taking the normal shove action (replacing one of your attacks) it just gives you and additional option. You could even do both in the same turn if you wanted (including shoving even before you attack).
Bonus actions are valuable
If you are playing a character that makes heavy use of their bonus actions for things other than Shield Master, then this part of Shield Master is probably not going to be the most useful to you. You only get one bonus action per turn and it is good to spend it wisely. However, not all characters do have a bonus action ability that they use commonly. In this case, if one wants to shove, this allows that character to put that bonus action to good use. In the end it all comes down to the specific character build you have/want to have.