Binance is a massive exchange, for it to remove DOGE/ETH is a very big deal and a big blow for DOGE.
They still have the DOGE/BTC pair so still good for those traders.
We don't know the reason why they delisted it but they have their own reasons for doing so.
This is one of the old coins that I still use today for transfer purposes because of low tx fees and you can find it even in small exchanges.
However, there are exchanges that are charging high number of doges on their withdrawal fees but still low compared to btc fees.
I'm the same as you as I mostly use Doge when it comes to exchanges as the fees are a lot cheaper but like you I have also seen some exchanges charge a fair bit for Doge. I know one exchange that charges 25 Doge for withdrawals but its still cheaper then BTC fees.
Yobit charges 500 doges for a withdrawal!!!!