No, Binance didn't get big because of Tether or leverage. It was already eating most of the crypto-only exchanges volume when altcoins were almost exclusively paired with BTC and before it acquired its perpetual futures business.
The main reason they managed to attract so many users is flamboyant altcoin pump and dumps orchestrated with ICO teams and Chinese whales, and pioneering exchange coins with the securities/utility token mash-up called BNB. Users came to chase the pumps, and stayed when they were invested in the platform itself.
Binance's thriving in its later phase even as Coinbase ramped up its altcoin listings can certainly be attributed to Tether though.