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.

Once per 1st level or higher spell.

When you cast an illusion spell of 1st level or higher, you can choose one inanimate, nonmagical object that is part of the illusion and make that object real.

The rest of the paragraph simply says when you can do this "on your turn as a bonus action while the spell is ongoing"; not that you can choose a new object each turn. All this does is mean that you do not have to make an object real at the start of the spell, you can do this on any of your turns (including the one where you cast the spell) as a bonus action. When you do this you choose the object and it lasts 1 minute but you only get one real object per casting.

Minor Illusion is not a first level or higher spell so that won't work for you. The ultimate limitation is the number of spell slots you have available to cast 1st level or higher illusion spells plus any magical items that allow you to cast such spells (items that produce illusions without specifically saying "you cast" don't benefit from this). The practical limitation is as long as this is the most effective use of your action.