Short rests require an hour in which
a character does nothing more strenuous than eating, drinking, reading, and tending to wounds.
(PHB, page 186.)
Eldritch Master takes one minute, and doesn't specify what you can and can't be doing in that time. It's unclear (i.e. open to DM ruling) if you have to spend that minute doing nothing except entreating your patron, but even if that's true, sometimes, you don't have an hour. If you're running away from something, you might just be able to get a minute lead on it to quickly get your spells back. An hour lead is a lot less likely.
A minute is a very short time out of combat, and even in combat, it's 10 rounds. In a protracted fight you could probably pull this off. In a really protracted fight, you might need to. For example, if you're fighting the Tarrasque, your Eldritch Blast will be entirely useless. Once you're out of spell slots, you're going to be standing around doing nothing. (Slight exaggeration.) Your allies might be able to keep you alive for 10 rounds so that you can get some spells which have a chance of achieving something.
It shouldn't need pointing out, but an hour is 600 rounds, which is just not going to be possible in any combat. (Unless you're fighting a Dire Half-Dragon Fiendish Snail or something, but that's just getting silly.)
Of course, all of this is somewhat pointless if your DM is ok with you doing other things during the minute it takes to use this. For example, even if you're only allowed to use your movement, running around for a minute screaming at your patron for help seems completely achievable. If your DM rules that you can fight as normal during that minute, then this feature might actually be too powerful. Get into a fight with a tough enemy? Start praying to your patron. That gives you 10 rounds to use all your resources, at the end of which you just get them back immediately. Personally, I wouldn't be quite that lenient, but that's up to your DM.
No, you do not need to know fireball - the language of "you can cast" enables you to cast it through this feature regardless of whether or not you know the spell. This invocation uses the same language as the ones that allow you to cast spells the warlock usually cannot learn, such as Dreadful Word or Ascendant Step.
To supplement this answer, note that it specifically has a spell known prerequisite of only eldritch blast, not eldritch blast and fireball. It would likely state the latter if it required that you knew fireball in order to make use of it.
Best Answer
Yep, it's useless
The text of the Eldritch Invocation is pretty clear...you can cast bane using a Warlock Spell Slot.
If you don't have Warlock Spell Slots, you cannot cast it. This same rule applies to a multi-classed Warlock: they have to use a Warlock spell slot for this, they can't use a normal spell slot that they acquired from, say, Multi-classing Sorcerer.
The language used across these Eldritch Invocations is very consistent. If it was intended that you could cast these spells using some other Spell Slot, then it wouldn't specify that you had to use a Warlock Spell Slot. It would instead say something like...
But it doesn't. It would have been very easy for them to leave the specificity out, and they did not. Thus it's a safe bet that this is intentional.
This is supported by an interview with Jeremy Crawford (lead rules designer for WotC), found here. To transcribe...
While not a definitive ruling (as it's just an interview, not a formal Sage Advice post), and is using Crawford's usual circuitious language...I find this to be fairly strong support. Crawford appears to consider "you need a Warlock Spell Slot to cast this" to be a prerequisite.