Assuming, as you've requested, that "that's boring, don't track it" is off the table, then yes, your intuition is right: this is a logistics problem. And to the contrary, it is the bread and butter of many groups' style of playing D&D, with a tradition stretching back to the very first days of the hobby in the early 1970s. Liking these ideas and wanting to involve them in your roleplaying is not in vogue now, but it is certainly within the range of acceptable ways of playing D&D. (I personally quite like the challenges and "putting myself into their head" feel of playing in this way.)

And, wonderfully, D&D 5e is flexibly designed in such a way that it can easily be played in this way without fighting the game.

If it being a logistics problem is valid for a group's approach to the game, then it's one of the fundamental puzzles that the players are responsible for solving, and doing so requires accounting for where everything goes, both in the sense of "accounting" as keeping track of things, and in the sense of justifying themselves.

When you're playing the game like this, you appeal to real-world knowledge, except insofar as the game provides replacement rules or guidelines that replace your real-world knowledge. (All RPG rules work like this — replacing or modifying your default knowledge of how things work — but that's especially true for managing mundane issues like this.)

So, with that preamble out of the way, where does all that stuff go?

Accounting for your stuff

A backpack only holds so much, yes. As you can in real life though, you can strap stuff to the outside, hang stuff off the bottom, and carry things in your hands while walking. That absurd hunting trap can go anywhere you can justify, and there are as many ways to solve the question of "where do I pack this?" as there are people with different gear-packing strategies. I'd hang it off the bottom of the pack, myself, but maybe you have other things that would better go there? Just like packing for a real hiking trip, things get juggled around until it all just works — or you give up on an item as too much bother to take with you.

Once you figure out a place for everything and everything's in its place, you've solved the problem of "how" naturalistically.

Swords, shields, ammo, and clothes

Indeed, items of combat equipment don't go in a backpack. Or they could, I guess, but that's an inconvenient place for them. Scabbards let you carry a sword on your belt with ease, and also make it quick to wield. A quiver does the same for ammo. These free up your backpack and hands for other gear. Shields get strapped to whatever they can be solidly strapped to, such as over the backpack or hung off the saddle.

Quivers (for arrows) and cases (for crossbow bolts) are listed in the equipment tables with prices and weights, so they don't come with the ammo and you have to buy one, and they aren't weightless. (A note to the less logistics-minded readers: without "a quiver, case, or other container" for your ammo, you don't get the benefit of the Ammunition weapon property's free reloading rules.)

Scabbards are not listed in the equipment chapter. Since swords on the smaller side are fairly useless without somewhere to keep them, it's probably safe to assume they come with scabbards when you buy them. (I personally find this a curious oversight, especially since quivers and cases are detailed, and other editions with "you want logistics? you can do logistics" levels of equipment detail do list scabbards. You may wish to add an entry for scabbards to the equipment lists your group uses. After all, when you find a bare blade in a dragon's hoard, you're going to want to buy a nice scabbard for it, right?)

Most armour, including plate harness, was worn over some kind of clothing. You might want a spare change in your pack, or you can just be grubby between the baths and laundry service provided at the inns you stop at. Medieval-level societies were often pretty filthy by default, so your character has a range of legitimate/believable choices in the realm of hygiene.

Sharing the burden

And speaking of saddles, when you've got too much gear to conveniently carry, that's where a pack horse, donkey, or riding horse comes in. (People who can't afford a beast to carry their stuff would dump it in a hand-cart instead — and though that's not a great way to travel long distances with all your possessions, people did what they had to, and sometimes travelling a long way with all your possessions in a hand cart is what people had to do. Hopefully your adventurer is more financially successful by the time they have to haul more than they can personally carry, though, and won't be reduced to such a sorry state.)

Carrying the loot

If you're well-equipped, you're probably already pretty burdened just with "necessary" gear. That doesn't leave much room for extra stuff you find in your travels, like piles of gold that can buy you that mule back in town. Old-school D&D players have long since figured out the solution to this: the sack. Sacks are lightweight, small to pack when empty, can carry literal sacks of gold, and can be carried one in each hand while travelling, leaving your packed pack unmolested by the storage needs of the new stuff. If only lightly filled, a sack can be tied to a belt, and carried even more easily.

The humble sack is a treasure-hunter's best friend, even ranking before the mule to carry even more sacks.

Less amazing than the sack, but still useful, is the belt pouch. Easily carried on the belt (or belts, if you opt to wear a second one over a shoulder, crossing your chest), pouches add a small amount of carrying space at no extra inconvenience for managing them.

Isn't this all unwieldy when actually, y'know, adventuring?

Yes, it is. Fighting while holding two sacks full of dragon's gold is not really going to end well. Or begin well, really. When combat looms, the traditional response (here, tradition referring to the grognardiest RPGers who have been playing this logistics-heavy way for decades and still are) is just as simple: drop your sacks and maybe even your pack to reduce your personal encumbrance, and then have at them with sword, bow, and spell.

Then, pray you don't have to run away — or if you do, you have a chance to snatch your treasure and gear back up!

A pack animal makes this easier yet: drop their lead when combat looms, and advance far enough forward that the melee doesn't come near the animal and spook them. (Though, tracking down a spooked animal with all your loot and gear on it can be an interesting adventure all its own, and lead you to places you might otherwise have never found!)

Better yet, hire a handler to hold your animal's lead while you, their master, engages the enemy in combat. Obviously you have to pay them, but the convenience of knowing your animal isn't going to run off unattended is worth paying for. Of course, that's assuming that they're loyal henchmen, but that's why you pay well, maintain a good reputation, get to know them at a personal level, and learn leadership skills that inspire people to be loyal followers rather than double-dealing traitors.

Plus, the risks and rewards of having and handling hirelings adds another social dimension to the game, which adds interest to play, and provides for another direction for complications and opportunities to arrive and convolve your gameplay. The more ingredients going into a roleplaying situation, the more engaging and emergent the situations that result. Just like a straying pack animal can lead to adventure, so can friendship and friction with a servant. But how the mundane can convolve play situations into deeply-engaging adventure is itself a whole other subject...

There are not rules for riding with a non-military saddle vs riding bareback, though that would be within the rights of the DM to rule as such (the rules are currently silent on the difference, and only provide an explanation that a military saddle provides advantage on checks to stay mounted).

Dash is one of the normal action types for a nonintelligent mount, so I would definitely not make you make a check in that case unless another circumstance arose.

There are effects that would dismount you. The most common is an effect that moves your mount against it's will (DC 10 Dex save).

The rules for mounts are outlined in two places p198 and p155. It would be good to review these two pages to get the full picture on what is and is not codified currently about mounts.