The incentive is not just that you move faster. It's that when you move faster, you have fewer random encounters.
Encumbrance is only relevant when you have random encounters. And random encounters pose a serious risk.
Encumbrance is just one mechanic of a system of something like 10 different mechanics that all rely on each other to work. It's just that in recent games using the D&D label, that system no longer exist and you're just left with encumbrance existing all by itself in a void, serving no function.
I want to iterate on this, because I think there's something missing here - it's not
just about avoiding random encounters. There's also a subtle push-your-luck play going on here!
Why are you encumbered, and therefore moving more slowly, and therefore at greater risk of random encounters and the risks they entail? Because you're carrying more
stuff - most importantly, provisions and loot.
The reason encumbrance matters in classical dungeon-crawling play is because
it is the source of the crucial gameplay decisions that players have to make in that style of play: are you willing to risk more random encounters in order to be better equipped against the other kinds of dangers the dungeon presents? How much loot are you going to try to take with you? Are you willing to risk more random encounters on the way back out so that you can haul more loot? Or are you instead going to be
less prepared, and haul out
less loot (perhaps a disappointingly lesser amount) in order to ablate the random encounter risks?
Because the point of classic play is to collect loot and bring it out of the dungeon, the encumbrance mechanic and its implications set up the push-your-luck play: you want to carry as much loot as you can, so maybe you push your luck and encumber yourself. The greater risk increases the tension and emotional stakes - and hence, the fun!
(Parenthetically, I should note that nothing about encumbrance driving the above play decisions requires that it be, say, pound-weight or coin-weight encumbrance! Even many OSR dungeon-crawlers these days have abandoned pound-weight or coin-weight measurements in favour of some other way of tracking encumbrance.)
Contrast this with heroic adventure play. In this kind of play, the
risk versus reward balancing act that players have to consider is something completely different, and may well not have much to do with earning loot at all. It's probably more closely related to things like player character goals - what do you want, and what are you willing to give up to get it? - or heroic dilemmas - if you can't save everyone, who do you save? - or heroic quests - can you make it to the Temple of Doom in time to stop the conjuring of the Demon of Doom, and how will the actions you take increase or decrease your chances of success?
In heroic play, "how much can you carry on an ongoing basis?" is usually just not an interesting question with enjoyable gameplay coming out of it, and I don't think adding the complexity of a widget that makes tracking encumbrance worthwhile on a stand-alone basis is adding any depth to the actual focus gameplay.
If anything, the answer to encumbrance in modern D&D (arrant nonsense such as "games using the D&D label" notwithstanding) is to jettison it. If you absolutely
have to include some kind of limitation in order to appeal to your own or your players' preferences as regards verisimilitude, use a simpler mechanic, always ensure the player characters have plentiful ways within the in-game fiction to haul piles of gear around, or, as suggested by
@Crimson Longinus, just kind of handwave it as long as no one is trying to game the system.
(That being said, having read through the blog post linked in the OP, it strikes me that the game the original blogger is running is the kind of game where "how much can you carry on an ongoing basis?" - or perhaps, better worded "how valuable do you find it to over-burden yourself with
stuff during any given phase of the adventure?" is a more interesting question that is actually driving gameplay! So it certainly makes sense that they want to have an encumbrance system with teeth - just different teeth, so it seems, than would have been biting on the player characters in classic D&D dungeon-crawling.)