The challenge isn't building a website that can stream video. The challenge is streaming 15% (I think it used to be even higher) of total internet traffic, which is what Netflix does. iPlayer shows the ability to do the former, not the latter.
This isn't to say that BBC couldn't have pulled it off, just that iPlayer isn't evidence that they would have.
iPlayer streams about 350 million shows a month. It accounts for a fair bit more than 15% of UK internet traffic and is more popular than Netflix within the UK. I know former staff who have disclosed privately some of the traffic numbers to me - they have the capability to do this sort of traffic, yes.
Whilst certainly not at Netflix scale, the BBC do have their own decentralised caching infrastructure called BIDI [1] that is frequently deployed inside large UK ISPs. This is similar to Netflix's OCA, Google's GGc or Facebook's FNA.
This isn't to say that BBC couldn't have pulled it off, just that iPlayer isn't evidence that they would have.