I don't see a very good argument here against allowing this. Yes, you would technically be flying with hover by walking. However, manipulating an object is an action. So, at best, each step would require a bonus action. That would give you a breakneck speed of one. Two, if you also used your full action for a second step. And, since at least one of your feet are immobilized between steps, you'd also be restrained the entire time you wore them.

All for the low, low cost of two third level spell slots. So sure, you could do it. I suppose there are times when the permanent version would help out if you don't have a better option. But, IMHO the permanent boots would make a great cursed item. Lots of comedy relief value for an NPC.

The existence of an option does not necessitate the existence of alternatives

The spell says you can move the object normally, but it doesn't say that you can not move it normally. Concluding from the fact that it gives you the option to do something that you must also have the option to do otherwise adds existential import to 'can' that is misplaced. Typically, when game designers want to concisely imply existential import for negative actions while defining a positive action the word they use is 'may', not 'can'.

That said, you can still let your allies do your weird walk/flying thing, it just takes one full minute per step:

You can also set a password that, when spoken within 5 feet of the object, suppresses this spell for 1 minute.

And also you may still be able to do this-- the spell certainly doesn't say you can't-- but you need to confirm your GM lets you choose to be affected by the spell first.

I think the issue here is that people are focusing on the 2nd part of the description of the spell and forgetting the first part.

I think, that as written the spell would allow you to walk through the air, if you cast it on both your shoes.

Immovable Object Says:

You touch an object that weighs no more than 10 pounds and cause it to become magically fixed in place.

This means you cast the spell and the object is fixed in place. Full stop.

You and the creatures you designate when you cast this spell can move the object normally.

This is a separate sentence. It offers the option of moving the object if you are the caster or one of the people he designates. It does not require him to do so. (I looked it up, grammatically can in this sentence is an option not a directive.)

I admire creativity, and as a DM I would allow it, but I would also use it against players by having my villains also use it. and judge falling prone very harshly. Dispel Magic would work well to cause falling damage. If tackled in the air, I might use the lingering injuries rule (DMG p272) For broken ankles. Because you have to willingly move the shoes. If someone applies a sideways motion to you and you are not choosing to move, the shoes would stay in place and potentially cause severe damage to your ankles or knees.

Unless you can fly normally, the spell does not allow you to fly.

Immovable object says:

You and the creatures you designate when you cast this spell can move the object normally.

I'm not sure where the idea of flying with these boots comes from. "Move the object normally" means that after casting the spell, you can move the object in any way you could move it before casting the spell. If you cannot fly before casting the spell, you cannot fly after the spell, as flying is not something you do normally.

No, the boots are always able to be moved by you

The spell says

You and the creatures you designate when you cast this spell can move the object normally.

The spell doesn't say you can tell the object to fix in place and then be free floating for you again. It is either fixed (for creatures not designated by you) or freely moveable by you (and those you nominate).

So you don't have the option to make it immovable for you and then movable again a second later. For you, it is as if the spell is not in effect.

It will probably not allow you to fly

You can check with your DM, but it seems pretty clear the spell is not supposed to do this, because that would be extremely powerful, and also completely the opposite of what it is supposed to be for (which is an object that doesn't move most of the time)

If I were DM, I would not allow this. If you wanna fly, cast Fly. If you cast Immobile Object on your own gear, I'm going to make you have a bad time. (Of course I'll warn you about this, first)