[RPG] Can an Arcane Trickster Rogue learn all of the wizard spells from the low levels it has slots for

My take on this is as follows:

1. Arcane Trickster level 17 class ability allows for stealing spells somebody is casting (this isn't restricted to wizard spells). Skill is located on PHB pg. 98, emphasis is mine.

Spell Thief

At 17th level, you gain the ability to magically
steal the knowledge of how to cast a spell from
another spellcaster.

Immediately after a creature casts a spell that targets
you or includes you in its area of effect, you can use your
reaction to force the creature to make a saving throw
with its spellcasting ability modifier. The DC equals
your spell save DC. On a failed save, you negate the
spell’s effect against you, and you steal the knowledge
of the spell if it is at least 1st level and of a level you can
cast (it doesn’t need to be a wizard spell). For the next
8 hours, you know the spell and can cast it using your
spell slots. The creature can’t cast that spell until the
8 hours have passed.

Once you use this feature, you can’t use it again until you finish a long rest.

2. This ability lets you know the spell and prepare it for up to 8 hours.
3. Spell progression is up to level 4 spells.

Now, this character requires a level of Wizard to make the spell stealing permanent. From this, it follows that:

Your Spellbook allows you to copy any wizard spell you know so long as you have the spell slots for it and you have the time to decipher and copy it (2 hours and 50 gp per level, conveniently meshing nicely with the 8 hour limitation on the spell you stole.) Emphasis to pertinent sections of the spellbook are mine (PHB pg. 94)

Your Spellbook

The spells that you add to your spellbook as you gain levels
reflect the arcane research you conduct on your own, as well
as intellectual breakthroughs you have had about the nature
of the multiverse. You might find other spells during your
adventures. You could discover a spell recorded on a scroll in
an evil wizard's chest, for example, or in a dusty tome in an
ancient library.

Copying a Spell into the Book. When you find a wizard spell
of 1st level or higher, you can add it to your spellbook if it is
of a level for which you have spell slots and if you can spare
the time to decipher and copy it.
Copying a spell into your spellbook involves reproducing
the basic form of the spell, then deciphering the unique
system of notation used by the wizard who wrote it. You
must practice the spell until you understand the sounds
or gestures required, then transcribe it into your spellbook
using your own notation.
For each level of the spell, the process takes 2 hours and
costs 50 gp. The cost represents material components you
expend as you experiment with the spell to master it, as well
as the fine inks you need to record it. Once you have spent
this time and money, you can prepare the spell just like your
other spells.

Note: I've used the portion on copying a spell into a spellbook because it specifically utilizes the terminology, "When you find a wizard spell….." Whether you steal it from a scroll, a spellbook, a magical device that imparts the knowledge, training from another wizard, or in this case, the mind of another wizard, so long as you as a character can cast the spell, you have "found" a spell.

1. This spell has to be a wizard spell to add it to your spellbook (this is specifically stated in the first sentence of the spellbook description for Copying a Spell into the Book.)
2. Additionally, since a wizard can write down whatever spells they currently have prepared in the event they lose their spellbook, this stands to reason that the entire spell is memorized. Since the Arcane Trickster steals the knowledge of the spell completely from the wizards mind, this also stands to reason that they would be able to reproduce the spell since it says they can directly in the ability.
3. Under the Spellcasting section of the PHB pg. 203, the requirements to cast a spell are neatly summed up and very explicit. Verbal, Somatic and Material components are the only three components that are potentially necessary. In order to pen a spell, you must know in what manner the physical components are used and train to memorize these components for each spell you are going to be utilizing. This is why wizards require extensive preparation. The Arcane Trickster falls under the preparation field in that the caster portion acts like a wizard. This is covered under the Arcane Trickster Spellcasting Ability portion on PHB pg. 98. Unlike a wizard, the AT only has a small number of spells, and thus knows them instead of preparing them. If you dip wizard though, you can expand the list of spells you know, as well as expand the list of spells you can prepare.
4. With that in mind, it stands to further reason that the written portion of a spell are the components and their specific interactions. That means when you steal a spell you have the components necessary to cast it firmly fixed in your mind for the next 8 hours and are able to replicate so long as you have spell slots. With the wizard's ability to record wizard spells (specific) that you have prepared, this closes the circle in that you have the spell firmly fixed in your mind, and can reproduce it. With that in mind, penning the requirements takes at maximum 8 hours (you only have up to level 4 spell slots), and this ability lasts for specifically 8 hours.

Absolute last point:

Functionally, this doesn't change the class or overpower it in any way. As a wizard at level 3, and Arcane Trickster at level 17, you can pen any level 2 or lower wizard spell already so long as you find it. I submit that due to this, stealing a spell from somebody's mind can act as finding a spell, and the character can utilize the multi-class dip to write it down accordingly.

So with that in mind, does the logic follow that so long as the Arcane Trickster takes a single level dip into Wizard they can write down any spell thief'ed wizard spells up to level 2?

If not, can you illustrate why?