I disagree. I have had to create house rules and my own content to get around these various issues. Here is a quick list of what I've done, and I'll be happy to go into more detail if anyone is interested:
My point was to change those things you don't need house rules. You can do it on the fly (at least that is what I find).
1. Broken CR system. I fixed this by completely re-writing every monster that I used in the campaign encounters. I took my characters and averaged their vital statistics: HP, Save Bonuses, Attack Bonuses, AC, Saving Throws, etc. I then balanced the monsters they faced based on the abilities of the characters. So if I wanted a monster to last for 4 rounds, I based its HP off of 6 or so average hits from the party. Its damage potential, I balanced it against the average AC and HP of the party. I have a feeling that the designers of 5E didn't even take this basic level of care in planning out the monsters.
The CR system works fine for a certain set of groups. It is fairly fast and simple and a good rule of thumb. Your approach, though more balanced for your group would have to be recalculated for every group and be adjusted each level as the PCs advance. And it still doesn't take into account player skill / group synergy. I just don't think that is realistic for a game. I've not played a lot of other games, but there is no edition of D&D that does this. I think you are asking to much. Now, I do have my own issues with CR and encounter design, but these are not them.
2. Poor selection of official adventures. I ran into this issue after the travesty that was Storm King's Thunder. When we had nothing left to play, I began writing my own adventure with the goal of publishing on the DM's Guild. We started playtesting. That's when I discovered that the CR system was broken (after 2 TPKs in three levels) and necessitated in all the changes I'm listing here.
I can't speak to official adventures as I don't play them. I know some think they are great and some think they are a mess. Regarding TPKs, I find this hard to understand as almost everyone who speaks up on these boards thinks 5e is D&D on easy mode. IF you go by the encounter guidelines and suggested CR you should almost never see a TPK. I think you a real outlier on this one.
3. Monsters need purpose. So I added some cool abilities to the monsters. Like the ability to push the characters away or grapple them and actually have a mechanical impact on the game (which isn't present in 5E). Others I gave dangerous ranged attacks (if the goal was to have an artillery "glass canon" creature) or the ability to soak damage and regenerate (if it was a bruiser). Monsters weren't designed in 5E to serve a tactical purpose. I changed that in my system. How do I know they weren't designed this way? I created a spreadsheet that attempted to classify creatures by level and role: is it a tough creature with a rock-solid AC, is it a high-damaging threat with an Achilles' Heel? The project had to be abandoned because there was no rhyme or reason to the design.
Pushing and grappling are standard combat actions that most , if not all, monsters can do RAW. You don't need all the rules tp be in the stat block. It is also completely within RAW to change weapons and armor and HP. You don't need to rewrite monsters to do any of these things. That being said, the monsters are, I agree, designed a little simple for my taste, and since I like making monsters I have posted over 200 custom monsters on these boards that do much of what your talking about. However, I truly believe that is a feature and not a bug in the system for 2 main reasons:
1) They work as is for a large group of people, most notably newcomers IMO
2) 5e puts a lot of weight on player and DM creativity - like older editions. You can do whatever you want - it just isn't spell out, but you do have a hose of guides to help you adjudicate situations. Some think this is a mistake, but having spent a lot of time playing and making monsters for 4e, I have come around (partially) to the 5e philosophy. All the time on the old WotC 4e boards people seemed constrained by the monster statblocks. They thought the monster could only do what is in the stat block - though not true; I think the origin of that viewpoint is the stat-block that tried to define everything.
4. Undercooked tactical combat. By adding in cool, flavorful abilities on the monsters, this really brought life to the combats. The party was facing down a warren of goblin slaves (minions), while duergar soldiers pushed and prodded them into the path of a rolling war wagon. The war wagon shot mortars of crowd controlling poison gas, obscuring the sight of the party as the duergar overlord fired a hand canon into their last known position. Dangerous zones, shoving enemies under the treads of a war machine, the heroic feeling of cutting down multitudes of opponents - these are all things that don't happen in 5e, which made numerous set piece battles that will be long remembered by my group.
see my answer to 3 above. I would just like to add you can have the same encounter you describe above in 5e.
5. Move and swing sword. By giving the monsters more flavor and more abilities, the DM can raise the stakes. Climbing the war machine to get out of the poison gas and kicking away the goblins who are trying to pull you off, just so you have a better shot at the leader with the hand canon, this is so much more exciting than just chopping away hit points. But when the design paradigm of 5e is "higher level creatures just do more HP damage and can take more HP damage," you get a sameness in the game.
I don't know why you would play 5e that way, we certainly do not. It seems more about how you play it than the system itself.
Wow, this is reminding me of the arguments I had when I was playing 4e trying to tell people they can play edition just like there whatever older edition. I really think it comes to to play style and not system.
6. Inspiration. I made an alternate system for inspiration, which allowed it to be used to pull off battle maneuvers (similar to Encounter powers in 4E).
Cool! I just allow them to make a check, which is RAW, that way it is tied to their talent / ability which I like better than inspiration (to be honest we don't use inspiration).