Yes, as larger states form they will be able to field larger and more effective forces, including fully armored knights and armored foot soldiers. Assuming a POD of roughly 1400 AD there aren't any such states in Europe, although the HRE comes close. After 1500, when the Spanish begin to expand, then we will see such forces appear.
A POD for the development of a repeating crossbow of the type you describe in Europe? I think that's what you mean, because with out it the repeater wouldn't be developed until after professional armies appear. In fact I don't see the weapon becoming useful until then any way, due to the massive amount of bolts needed to make it practical.
With regard to your tactical analysis, I agree, except that the attacker will close a little more rapidly than you seem to allow for. Starting at three hundred meters we can close to one hundred fifty in as little as thirty seconds while maintaining formation, which doesn't give much time for the crossbowmen to fire at them; perhaps enough for a couple of volleys. And if the crossbowmen withdraw to keep the range open they won't be able to reload; only the lightest of crossbows can be reloaded on the move. And the pikes have to move with them, or they become vulnerable to cavalry attack; such retrograde movements have a way of getting out of hand and resulting in the unit running off the field and out of the battle.
Here is my issue with that speed of closing. The fastest man in the world, Usain Bolt runs a 100 meter race in 9.58 seconds. If that speed is kept constant he would theoretically cover 150 meters in just under 15 seconds. A full out sprint is not conducive in any way to maintain formation. so lets put it down to a fast run, tack on anywhere from five to ten more seconds depending on how generous you want to be. Now put on a metal helmet, some light armor (perhaps leather), put a crossbow in his hands, and give him enough bolts for a meaningful number of shots. That easily exceeds thirty seconds, and expecting this performance along with sustained combat, is ridiculous. What's more, not all men in the unit are Usain Bolts, and not all men are even at the same level of speed, strength, and agility.
150 meters in 30 seconds is sprinting speed, not advancing in formation speed. Moving at such speed would make it impossible to maintain formation. It's hard enough to prevent unit brake up when soldiers are mounted on horseback, that's why cavalry typically advanced at a trot for as long as possible, and only charged the opposing line once they were close enough to limit the amount of formation brake up enough to keep a solid charging mass.
I'd give a well formed unit of men, 1 minute at the very least to cover that distance, and when you factor in the possibility of broken terrain, as well as other possible factors, I would expect typically more time to be spent. That gives at least two to six volleys time to be loosed at the advancing repeaters before they can even hope to return fire, and 50 more meters before they close to effective range.
What needs to be remembered is this is all assuming completely unopposed access towards said non-repeating crossbowmen. The beauty of the weapon is that it can be fired over fellow soldiers. Also as you pointed out crossbowmen will need pike escorts to protect them from exposure to cavalry attacks. This effects your advance just as much as my retreat. This fact means you've got to put out pikemen to shield your advancing crossbomen, and their armor, dense formation, and potentially unwieldy pikes, further slows any advance towards my crossbowmen by your repeaters.
Also in regards to the crossbowmen retreating out of range, your absolutely right that this movement could get out of hand, but it isn't like the repeaters are chasing the regulars down an empty field, there will be other soldiers to deal with while the crossbowmen reset and resume firing. Yes they can't reload on the run, but after losing a volley they can reload before they retreat, or they can retire after a volley and reload out of range. One idea I had was putting the crossbowmen out in front, firing until the repeaters could shoot back, and then falling back behind their pikemen and resuming their fire further away behind protection, possibly covered by crossbowmen already there. By sending some of my infantry forward I can also potentially move to prevent your repeaters from advancing within their 100 meter effective range, but still within the 150 meter effective range of my crossbows. Those fifty meters in my opinion are going to be the magic area where every battle along the lines of this scenario is decided: Either the crossbowmen deal out enough damage up to and in this area to turn the tide, or the repeaters are able to close the gap and strike back with enough force to carry the day.
I am glad we are mostly on the same page however, and I think your idea of a more powerful repeating crossbow then the Chinese one with a slightly slower rate of fire is a good one. I'm just saying it's going to need a significant amount of regular crossbows providing artillery support if it is to be an effective weapon. Like I previously mentioned, I'd think they'd be most effective mixed into a unit of regular crossbows as a significant though minority component, or mixed into units of regular infantry in order to provide organic firepower as ranges between armies close. A combination of both ideas might be even better.