Yes, but not a lot.
There may be more than this, but here are a couple of examples where the disease, potentially just once it sets in, can't be cured through the use of low-level magics.
Frigid Woe requires a specific cure.
Frigid Woe from Wildemount requires a specific antidote:
The only way a creature infected with the disease can be cured is by finding and drinking the manufactured antidote, a milky liquid stored in gold vials found in Eiselcross’s ruins.
This disease has a significant plothook vibe to it that matches "search for a cure": it progresses slowly but is inevitably lethal (barring extremely high saves) without the cure.
Aboleths have a disease that gets nasty after a minute.
Aboleths inflict the following on a hit:
If the target is a creature, it must succeed on a DC 14 Constitution saving throw or become diseased. The disease has no effect for 1 minute and can be removed by any magic that cures disease. After 1 minute, the diseased creature's skin becomes translucent and slimy, the creature can't regain hit points unless it is underwater, and the disease can be removed only by heal or another disease-curing spell of 6th level or higher. [...]
While it's still curable with spells after the set-in period, it sets a fairly high bar for the caliber of magic required to deal with it. It's not necessarily out of reach for a party fighting CR10 foes, but it's definitely a more pressing issue than a run-of-the-mill disease. Given how quickly it progresses after that initial minute, it's less of an adventure plothook, though.
It's more of a plot thing, though.
Diseases in 5e don't have all that many rules to them. The section on diseases includes a couple of sample diseases, but they're basically a character-impacting stat block. After the part you quoted, the DMG says the following:
A disease that does more than infect a few party members is primarily a plot disease. The rules help describe the effects of the disease and how it can be cursed, but the specifics of how a disease works aren't bound by a common set of rules. Diseases can affect any creature, and a given illness might or might not pass from one race or kind of creature to another. A plague might affect only constructs or undead, or sweep through a halfling neighborhood but leave other races untouched. What matters is the story you want to tell.
The complication of an outbreak is with regard to the plot. If the mayor's family grows sick with cackle fever but are quarantined, that's "a simple outbreak" that the party can resolve with "little more than a small drain on party resources," as they simply need four (or whatever) lesser restos, paladins, monks, etc. If someone stumbles into the Neverwinter River, fully blinded with Sight Rot, infecting hundreds by the time the party arrives, you've got "a more complicated outbreak" on your hands. Sure, Sight Rot can be cured with a lesser resto, but if you have hundreds of cases, growing by the day, your cleric is going to be tapped before you make a dent in things.