We had a new addition to our group and his first move was to purchase an elephant as a mount. All was well until we started a fight outside of the city and he began to use his elephant's charge attack in the fight. The PHB says mounts may only take dodge, disengage, or dash, so are mount's attacks not allowed?

Near the end of the fight, he dismounted the elephant and ordered it to attack while he also attacked. I got mad at the DM for letting him do this because I'm playing a Beast Master Ranger, so a Fighter getting an animal to attack with basically a free action felt off. The DM's defense was that the elephant would be no use once we get into dungeons and that my animal was going to level with me, but the mechanics of it being allowed were what irked me. They want to err on the side of reality rather than balance, but if so, then my animal (a baby bear that will level with me) will end up being incredibly op. How should trained animals in battle work in a way that doesn't destroy balance or walk all over the Beast Master's abilities?