The short answer:
Because technique (fingering/ways to move) and co-ordination are two different problems. If you start straightaway with hands together you will have all this problems at once, and will likely make many mistakes (more about that in a moment). By starting with hands separate, you can make sure that the movements and fingering are thoroughly ingrained before you move on to tackle the much more difficult problem of co-ordinating the hands.
On top of that, usually the left hand is weaker than the right. By going hands together straight away you never give the left hand a chance to develop and equal the right hand. In fact, your right hand will be limited by the left hand and working at a less than optimum speed. Meanwhile, the left hand, trying to keep up with the right will always be working in a state of fatigue. This is the perfect set-up for sloppy-sounding playing in the best case scenario and for an injury in the worst case scenario,
These are the obvious, superficial, reasons. But there is a deeper, hidden reason.
When you put hands together, in order to co-ordinate the hands, you will create system of cues and triggers where the movement of one hand triggers and cues the movement of the other. Most people are not even aware they are doing this (hence, deeper and hidden), but it is the only way you can play hands together. This system of cues will be deeply ingrained in your motor centres, and is commonly known as “hand memory”. Once something is ingrained in your hand memory, it will be there pretty much forever. If you go straight to hands together, even if it is a small passage, you will be making several mistakes (fingering, inappropriate movements – especially if you believe in slow practice – etc.). All these mistakes - which would be all right if done with hands separate – now get ingrained in your hand memory. By the time you realise what is going on, you will have to spend weeks to fix (sometimes it will not be possible) all these ingrained mistakes. So you will not save time, quite the opposite.
In fact, on the matter of time, people make the following fallacious calculation: “If I learn a piece with hands separate first, it will take three times longer to learn it than if I skip hands separate and go to hands together straight away”. This is fallacious because the real problem in playing the piano, is not playing the piano, but knowing your piece back to front. Learning it 3 times (RH,LH, HT) is going to provide you with the opportunity of a life time to truly dissect and learn your piece. So, not a bad idea at all
Now, instead of believing me, try this experiment: take two pieces of similar difficulty and learn one hands separate first, and the other hands together straightaway. If I am right, then the one you did with hands separate will be finished not only first, but it will sound much better.
Finally, you don’t need to learn a whole piece with hands separate before joining hands. It is perfectly all right to do that only on small passages (breaking a piece into smallest sections is a very sound idea), and when joining the passages to join them with hands together straight away. The only exception is counterpoint music (e.g. Bach) where you should learn not only the whole piece with separate hands but isolate each voice and learn each voice separately.
If you want more details, have a look at these threads where the subject has been discussed to exhaustion:
https://pianoforum.net/smf/index.php/topic,3085.msg27140.html#msg27140(Hands together: when and how – dropping notes)
https://pianoforum.net/smf/index.php/topic,4123.msg37829.html#msg37829(How to investigate the best movement pattern: Example Scarlatti sonata K70 – How to work out the best fingering. Example: CPE Bach Allegro in A – Slow x slow motion practice – HS x HT – practising for only 5 – 10 minutes)
https://pianoforum.net/smf/index.php/topic,3085.msg44855.html#msg44855(Hands together – dropping notes – when to learn HT and when to learn HS)
https://pianoforum.net/smf/index.php/topic,2998.msg26268.html#msg26268(Scales HT, why? – why and when to practise scales HS and HT – Pragmatical x logical way of teaching – analogy with aikido – list of piano techniques – DVORAK – realistic x sports martial arts – technique and how to acquire it by solving technical problems – Hanon and why it should be avoided - Lemmings)
https://pianoforum.net/smf/index.php/topic,3039.msg26525.html#msg26525(how big are your hands, and does it matter? 7 x 20 minutes – exercise/activities to strengthen the playing apparatus – ways to deal with wide chords – the myth that Richter was self-taught – 3 stages of learning – Example: Chopin militaire Polonaise - scientific principles for testing practice methods – Example: Prelude in F#m from WTC1 – when to join hands and why HS – practice is improvement – the principle of “easy” – Example: Chopin’s ballade no. 4 – repeated groups)
https://pianoforum.net/smf/index.php/topic,4797.msg45744.html#msg45744(No skipped steps – Bernhard enlightens further and tells the usual places where students go wrong – Ht x HS)
https://pianoforum.net/smf/index.php/topic,4858.msg46087.html#msg46087(Paul’s report on B’s method. Feedback from Bernhard including: HS x HT – Example: Lecuona’s malaguena – 7x20 – need to adjust and adapt – repeated note-groups – importance of HS – hand memory – 7 items only in consciousness – playing in automatic pilot - )
Best wishes,
Bernhard.