What counts as a 'dilemma'?

Could go either way.

What is a dilemma? Is it simply a problem one has (as in ‘we are facing the dilemma of dealing with the angry prescriptivists’)? Or must it be a choice between disagreeable alternatives (as in ‘we are facing the dilemma of dealing with the angry prescriptivists or the drunken linguists’)? Many current style and usage guides remain firmly of the opinion that a dilemma is only described by the latter of the preceding options.

Dilemma does not mean simply a problem; it means a choice between disagreeable alternatives.
— The New York Times Manual of Style and Usage, 2015

Despite the assurances of The New York Times Manual of Style and Usage, our records indicate that dilemma is quite often used in a sense that we define as “a difficult or persistent problem.”

Birdsall has done extensive research — this is the writer’s dilemma, to have unearthed so much great material, you can’t bear to leave any of it on the cutting-room floor.…
— Ligaya Mishan, The New York Times, 9 Oct. 2020

Hovering over the postponement or cancellation is a larger dilemma facing museums: how to account for growing demands for equity and representation on the gallery walls when the Covid crisis has shrunk budgets substantially.
—Julia Jacobs and Jason Farago, The New York Times, 25 Sept. 2020

The story crystallized a key dilemma of the 21st century. As authoritarian leaders oversee economic success, the importance of civil liberties — ensuring freedom from arbitrary punishment and harassment — may dwindle.
— Jochen Bittner, The New York Times, 22 Sept. 2020

Broad Use of 'Dilemma'

When dilemma first came into use in English, in the early 16th century, it was as a term of rhetoric, meaning “an argument presenting two or more equally conclusive alternatives against an opponent.” By the end of the 16th century the word had begun to broaden, with meanings such as “a usually undesirable or unpleasant choice” or “a situation involving such a choice.”

Dilemma, is a kinde of argument or reasoning, which euery way conuinceth him vnto whome it is spoken.
— Ludwig Lavater, Of ghostes and spirites walking by nyght and of strange noyses, 1572

Heere is a dilemma. Eyther put no holinesse therein and so followe no choyce of order: or else put holinesse therein, & therefore according vnto the premises you are iustly to be condemned.
— Edmund Campion, The great bragge and challenge of M. Champion, 1580

We begin to see dilemma being used in an even more broadened fashion, with the simple meaning of “a problem involving a difficult choice,” in the 18th century. Evidence of this is found in authors modifying dilemma with positive terms, such as happy and pleasant.

Mr. Walsingham being informed of my removal, the next morning he favoured me with a visit,—when I informed him concerning the pleasant dilemma of the preceding day, and the emancipation of Miss Laurence, who I presented to him.
— Ann Sheldon, Authentic and interesting memoirs of Miss Ann Sheldon, 1787

The great library and collection of natural curiosities which belong to the latter, chiefly engrossed his attention—afforded him utility and entertainment—but at the same time involved him in a pleasant dilemma.
— Dietrich Heinrich Stoever, The life of Sir Charles Linnæus, 1794

One great advantage arising from the multitude of these courts is, that their decisions are for the most part diametrically opposite to each others, the mind is by this means kept in a most happy dilemma, and remains in that salutary state of doubt….
— The Analectic Magazine (Philadelphia, PA), Jul. 1813

People have been complaining about the use of dilemma as a synonym of problem or predicament since the 1920s. Despite this century-long stream of admonitions (or perhaps because of it) the objurgated meaning has become, since the second half of the 20th century, the most commonly used sense of the word. Your use of the word in the sense of problem or predicament should not be a concern.