But: As fiat is only converted into BTC for a short time, and then converted back, would that drive the price up permanently and constantly? I guess you could argue that at any time xx remittances are needed in BTC.
Yes, that's why a quantity-theory based estimation would be needed.
Here an user tried to do that for the current price. According to him, Bitcoin is about 10 times overvalued. Unfortunately I didn't fully understand his math, but basically he says, "the more transactions, the higher the value".
I could imagine there is another way to find out the potential price if $500 billion in remittances were added (maybe the model is stupid, but it looks logical - although oversimplified - for me):
At this moment, we have
an estimated transaction volume of about 550.000.000 USD per day. That's about 200 billion USD per year.
We can simply assume: 200 billion$/year is the correct volume for a $2700 price and a $44 billion market cap. Every billion$/year of transaction volume would be then equivalent of a Bitcoin market cap of $220 million.
If we add the 500 billion$/year transaction volume (assuming BTC captures the total remittance market), that would mean the Bitcoin market cap would be about 3,5 times higher than today (200bn + 500bn = 700 * $220m = 154 billion = bitcoin price as of current coin supply: 9450$).
That is less than I expected but it shows the price, only because of remittances,
could well rise.