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.
Spells don't take on a higher level until you cast them using a higher level spell slot.
From page 201 of the PHB, under "Casting a Spell at a Higher Level" (emphasis mine):
When a spellcaster casts a spell using a slot that is of a higher level than the spell, the spell assumes the higher level for that casting.
You are not casting the spell when you pick it for your Arcanum so no, you wouldn't be able to select Mass Suggestion as a 9th-level spell when you reach 17th level.
Rules designer Jeremy Crawford has also unofficially confirmed on Twitter that unless the DM allows it, you can't choose higher-level versions of lower-level spells for Mystic Arcanum (e.g. choosing a 9th-level version of a 6th-level spell at warlock level 17).
Brendan Smith - @spaceysnorlax
@JeremyECrawford Can warlocks learn lower level spells as mystic arcana (eg circle of death as 7th or even dominate person as 6th)?
Jeremy Crawford - @JeremyECrawford
@spaceysnorlax As written, Mystic Arcanum doesn't allow spells of lower levels, but a DM could certainly allow it.
Best Answer
Per the Player's Handbook:
The Slot Level column never increases above "5th", so a warlock can never know a warlock spell of 6th level or higher.
Therefore the Mystic Arcanum spells must be known separately from the warlock's Pact Magic spells.