Here's one of the considerations I've been thinking of when it comes to balance:
Would I take this invocation if it worked the way I'm considering? (Assuming I wanted the spell.) And would I feel like I was taking a real hit to my effectiveness just to fit my theme?
That's pretty much the litmus test.
I like the suggestion for most of them of adding it to the spells known, and allowing it to be cast 1/day at the lowest level without a spell slot.
For polymorph and conjure elemental I understand how that might potentially be too good. It would allow you to basically keep one of your companions permanently polymorphed if you were going into a situation you could keep taking short rests in (including your typical wilderness travel, or old school killer dungeon). Conjure elemental takes a minute to cast, so you need to do it out of combat, but you could keep refreshing it such that with the two or three expected short rests you could have it up pretty much all day. Are those problems? A wizard can use planar binding to keep a conjure elemental going, so it doesn't seem too out of line. If a character is more effective being constantly polymorphed than being in their normal form...well that's sort of a problem that ideally needs a better solution than fixing some warlock invocations, but I still need to keep it in mind as a possibility.
But when I ask my self "would I take it?" with just turning those spells into 1/long rest without a spell slot, I find myself having that feeling that I wouldn't be happy with it. If it really fit a theme I was going for, I might take it, but I'd feel like I'm getting ripped off anyway. That's not how it should work.
So at what point does it feel worth it? 2/long rest without a spell slot seems like it might be a bit much. 1/short rest with a spell slot is a compromise between adding it to the list but not allowing all your slots to go to it, and it would be fine normally, but in a situation of actual "abuse" (like going into a classic killer dungeon where you really can stop to short rest every couple of rooms), it wouldn't do much to head it off. That's assuming "abuse" of it would actually be a problem, of course.
2 or 3 times per long rest with a spell slot seems like it might be the best balance point, but I'm not sure. It's kind of messy. I'd probably be satisfied with it as a player though.
Most of those are decent spells so I just let players ignore the "once per day" part and add them to spells known. Once per day without a slot beefs the class up too much for my liking. Those shift drastically towards really good at that point.