I don't think small private landlords will dry up either, plenty of people who have the cash to buy will still do so, offsetting the stamp duty against long term potential raises.
What will stop is people who release equity in their own home to put a deposit down on a BTL property, and essentially get the tennant to pay the mortgage. These kind of Landlords are actually dangerous in the market, they often pay interest only mortgages and are currently able to claim the tax relief, and rely on price increases to deflate the value of their debt.
Sadly, unless they are good landlords like the majority on here claim to be, they are often ill equiped to deal with the needs, they take long periods of time over repair (DS was a private renter for years and had more issues with small timers than big firms).
They compete with FTB for the flats that they can buy, and then end up letting to them and to be quite frank are only able to do so because they are subsidised, get rid of it, asap.
As I've said above big firms tend to buy higher value flats, especially in London, rather than the kind of starter thing a FTB could buy because of the huge returns you can get on them.
For example, someone I know who has been a property developer and landlord for eons, would say that you would only invest in something that would give you both a high yield and would certainly increase in value. To that end his firm now only buys properties in high demand areas that they can charge exorbitant rents on. His latest purchase is a place in an old gin factory in Camden, where he is able to charge his tennants about about £8.5 a month in rent for a furnished flat. This covers the huge service charge too!
Now there would have been a significant cash deposit, and his firm will get fairly low interest rate. It means that in less than 15 years (probably less with rent increases) the entire cash outlay on the flat will have been covered, and the assett will have appreciated in value. That is the only sort of deal that the big timers look at, no point piddling around with something that is going to pay you £1,500 a month rental, but cost you £350,000 to buy, which would take you 233 years to be in profit!