I've used Coinbase, Binance and Bittrex for over a year and have never had a problem with any of them. My only beef with Binance is that they lock withdrawals for some coins to control the rate of transactions and as you saw for yourself, their support is lacking. Bittrex is solid. I've only had to contact them once and they resolved my problem quickly. My only disappointment with Bittrex is they don't have a trading app but their site is mobile friendly.
RE: Best ways to buy Bitcoin without an ID