After catching up, I think the best use of my next battle master will be using push attack to launch characters across chasms they cannot hope to jump across on their own. 
For you, yes. For me, it depends on the type of campaign I'm running and more often than not I'm looking at tropes beyond typical human ability. For every table out there, no. That's part of the point -- the DM decides the style of that aspect of the campaign. Not a list of rules.
The difference is that I don't need a hard coded rule to make that happen in a campaign I want to run like that. That also doesn't mean non-magical abilities are going to match magical abilities for two reasons: 1) it's a limited vs limited resource; and 2) mundane might push the boundaries of physics but magic defies them by definition.
It's possible to turn such non-magical actions into limited resources to match the first point but the second becomes harder and harder to maintain a reasonable suspension of disbelief the more physics are pushed so the limits on magic in 5e are considerations to the limits on non-magic. It seems silly to do more without magic than with it but not vice-versa.
I'm very much inclined to allow an improvised reaction that would look heroic without magic. The key words being improvised reaction. I don't need a list to allow such a thing.
Give the BMX Bandit riot gear and an assault rifle and it's a closer analogy. He still cannot defy gravity to make that jump but the implication 5e martial classes would be useless in the fights in comparison to spell casters isn't valid.
I responded to this argument earlier. The skill list doesn't create equality in any given skill. Not all skill proficiencies apply to all classes and classes generally only have some proficiencies at all. There are opportunity cost, lack of proficiencies, ability score differences, and class attributes that change those proficiencies or ability checks going from one class to the next.
The scale of ability from those checks is also unrelated to the existence of checks for other classes.
What was the build? Knowing what made the bard effective and at what could be an interesting point of discussion.
Just don't blow off the request with speculation or generic comments. I'm asking because I want to actually see the bard build.
The help action is abused more than either of those, IME. But helping isn't magic so it's not perceived abuse.
I was that stuck-up PC. Launching a fireball into a window from which arrows are being fired is not always the appropriate response. The initial goal of stopping myself from getting shot was rather overshadowed by all the helpless people we failed to rescue by burning them to a crisp.
Clearly I was not as wise or intelligent and the character I was playing. Of course, that was many editions ago. I'm clearly a genius who never makes mistakes now.
I do sometimes in some campaigns and not others. I thought that was the point in leaving those checks in the hands of the DM.
The thing here is I believe it's possible to eliminate the spell lists. It's not like there aren't other ways to represent magic in a role-playing game. The challenge is in whether the game feels like D&D. The style of spellcasting originated from something rather unique and even though it's changed those roots are still a rather deep part of the feel.

For me the act of either lunging after and preventing or breaking an allies fall is very much classic heroic fiction
For you, yes. For me, it depends on the type of campaign I'm running and more often than not I'm looking at tropes beyond typical human ability. For every table out there, no. That's part of the point -- the DM decides the style of that aspect of the campaign. Not a list of rules.
The difference is that I don't need a hard coded rule to make that happen in a campaign I want to run like that. That also doesn't mean non-magical abilities are going to match magical abilities for two reasons: 1) it's a limited vs limited resource; and 2) mundane might push the boundaries of physics but magic defies them by definition.
It's possible to turn such non-magical actions into limited resources to match the first point but the second becomes harder and harder to maintain a reasonable suspension of disbelief the more physics are pushed so the limits on magic in 5e are considerations to the limits on non-magic. It seems silly to do more without magic than with it but not vice-versa.

I'm very much inclined to allow an improvised reaction that would look heroic without magic. The key words being improvised reaction. I don't need a list to allow such a thing.

That BMX biker wheelie is really amazing ... not much like teleporting the party escape out of danger or even like preventing the entire party from falling in a pit of doom as a reaction.
Give the BMX Bandit riot gear and an assault rifle and it's a closer analogy. He still cannot defy gravity to make that jump but the implication 5e martial classes would be useless in the fights in comparison to spell casters isn't valid.

Better warn the players its going to be an all diplomacy game so they can choose appropriately and the spell casters have skills too oh my things they can just choose and have happen combined with the awesome and eratic power to improvise or just be told no sounds like more versatility not less.
I responded to this argument earlier. The skill list doesn't create equality in any given skill. Not all skill proficiencies apply to all classes and classes generally only have some proficiencies at all. There are opportunity cost, lack of proficiencies, ability score differences, and class attributes that change those proficiencies or ability checks going from one class to the next.
The scale of ability from those checks is also unrelated to the existence of checks for other classes.
One of the groups I have watched has a bard massive massive effectiveness. What is the abuse?
What was the build? Knowing what made the bard effective and at what could be an interesting point of discussion.
Just don't blow off the request with speculation or generic comments. I'm asking because I want to actually see the bard build.

Right, I should have added the word perceived. Perceived abuse.
I'm not in the camp that has issues with Guidance and Bardic Inspiration.
The help action is abused more than either of those, IME. But helping isn't magic so it's not perceived abuse.

Let us not make this situational as it can go on forever right? I can just as easily throw some captive elves in there who drown because some stuck-up PC didn't want to get their sandals dirty.
I was that stuck-up PC. Launching a fireball into a window from which arrows are being fired is not always the appropriate response. The initial goal of stopping myself from getting shot was rather overshadowed by all the helpless people we failed to rescue by burning them to a crisp.
Clearly I was not as wise or intelligent and the character I was playing. Of course, that was many editions ago. I'm clearly a genius who never makes mistakes now.

I don't even allow Fast and Furious level of overcoming physics in my game without magic most of the time.
I do sometimes in some campaigns and not others. I thought that was the point in leaving those checks in the hands of the DM.
Try removing the list of spells and get back to me. It won't feel like D&D and for a reason. The Polearms were a 1e joke.
The thing here is I believe it's possible to eliminate the spell lists. It's not like there aren't other ways to represent magic in a role-playing game. The challenge is in whether the game feels like D&D. The style of spellcasting originated from something rather unique and even though it's changed those roots are still a rather deep part of the feel.