Does immunity to Exhaustion generally gives immunity to forced sleep?

There's only 1 source that I could find that makes this explicit: the Sleep spell, which explicitly mentions who is immune to it.
However, Symbol and Eyebite don't mention any immunity, and the Adult Brass Dragon doesn't mention any immunity either, despite all of these being called "Sleep".

In the case of elves, it's pretty clear cut, they have the "Trance" trait which gives them blanket immunity against all forms of magical sleep (dragon breath aren't magical effects, though). Warforged also have the similar "Sentry's Rest".
But for the Undead, the idea that Immune to Exhaustion = Immune to Sleep is only ever mentioned as a limitation of the Sleep spell. It is not a general rule. For everything else, it's the Unconscious condition they would need immunity to, which most Undead don't have.

This feels like another case of "magical darkness" where it feels like they should all block Darkvision, but really only the Darkness and Maddening Darkness spells do. 
Unless I missed something?

Exhaustion immunity does not give immunity to forced/magical sleep - it's a specific caveat of the Sleep spell.

Exhaustion as a condition is a measure of fatigue, which can be caused by lack of sleep, but can also be caused by things like environmental hazards and starvation. Immunity to Exhaustion just means that you shirk those factors, so Undead aren't worn down by lack of food, extreme temperatures, etc.

Likewise, species like Warforged and Elves are still susceptible to the effects of Exhaustion, even though they can't be forced to sleep. The two are unconnected.

Note that by strict RAW elves and warforged are not immune to the "sleep" and "asleep" effects of the Symbol and Eyebite spells.

I disagree. The name of these spell effects should be enough to constitute magical sleep. And it wouldn't be the first rule based on an effect's name (for example, the rule against stacking the same spell effect like Bless is based on the name being identical).

Immunity to exhaustion only gives immunity to sleep effects if the sleep effect specifies that creatures immune to exhaustion are immune. There's a plausible argument that most creatures immune to exhaustion should be immune to sleep, but by RAW they aren't.

However, I'd agree that immunity to sleep covers effects named sleep.

Quite frankly I think most Undead are immune to enough things already... 

Quote from Natrel >>

I disagree. The name of these spell effects should be enough to constitute magical sleep.

and also . . .

Quote from Pantagruel666 >>

However, I'd agree that immunity to sleep covers effects named sleep.

This take is understandable but it's not how the rules work.  The name of a feature has no mechanical consequence.  Features do what they say -- what they say is defined by their description, not their name.

Mechanically, even the Sleep spell doesn't actually put creatures to sleep.  The only reason why creatures such as Elves are immune to the effects of the Sleep spell is because the description of the Sleep spell explicitly says so:

"Creatures that don’t sleep, such as elves, or that have Immunity to the Exhaustion condition automatically succeed on saves against this spell."

Quote from up2ng >>

Mechanically, even the Sleep spell doesn't actually put creatures to sleep.  The only reason why creatures such as Elves are immune to the effects of the Sleep spell is because the description of the Sleep spell explicitly says so:

Sleep is not a defined term in 5e, and thus 'magic can’t put you to sleep' means that any effect that the DM rules is (a) magical, and (b) sleep, will be stopped.