Which baitcasting reels work well and are durable, meaning they don’t break easily? I don’t want reels that have to be handled with kid gloves. 

Reels with aluminum frames. 

Most Shimano and Daiwas 

My Lew's hold up very well, but so do my Qunatum's, and my Shimano's........lol

I'm not sure you're aware that the market is flooded with quality reels. Price range would give us a better idea on how to help you. 

Anything but Shimano....

You're going to have to look into reels at the $150 plus price point for low profiles. Otherwise, look into the Abu Ambassadeur C3's

I have abus that i bought and buy used over 50 yrs old that keep on goin and goin  ...you can find them all day from 30$ to 50$ on the bay looks are deceiving

Any reel $150 and higher will do what you need.

lew lfs ,abu revo,diawa fuego,shimano slx, and fpluger offer durable reels in the 100 market all these reels are good durable reels its up to you ultimately what your gonna pick.