What TMKIWI said.If you sort of picture working a hand water pump slowly through the stroke of suction and push then work it quickly so the push/pull overlaps it's much easier-for lack of a better way to illustrate it.
As for 2t street bikes,of course they did.I had two RZ350's.They were known in their day as giant killers and worked very well.Yamaha also sold the RZ500,a V4 2t street,Honda the NSR400 triple and Suzuki had the RG500 Gamma,a square 4 2t.
Back then,or now for that matter,I'd kill for a Gamma-drool,drool.
All these were sold in Canada,I don't know about other markets.
Oops,forgot one thing.The 2t dirt bike engine we presently have is a fairly inefficient design as well.As you've no doubt noticed,a YZ250 doesn't have twice the HP of a YZ250F,despite halving half the firing cycle.