The pet does decent damage. At level 3, it deals an average of 7 with an attack, with a chance of somethign extra. That's really not too bad at level 3. It matches the damage from a bonus-action attack as well as the damage from two hits while using Hunter's Marks. That's assuming we're talking about different options while being a Beastmaster. The Hunter does better damage thanks to Hunter's Prey options at level 3. This becomes less remarkable as we level up to 10, and start finding magic items, get feats that increase damage. A crossbow Expert with a +1 weapon would be dealing slightly more damage than the pet. Lets not get into a Crossbow Expert with Sharpshooter plus the damage from something like Collosus Slayer. The magic weapon also has a benefit of bypassing a lot of damage reduction, whereas the pet does not.
At level 11, the pet gets a significant jump in power - two attacks from the pet on top of the two from your base class. This doubles its attacks as well as potential damage output. However, the number of spells we can use hasn't gone down. Hunter Marks, Ensnaring Strikes, Lightning Arrow. If anything,we can use more of them, and each does more damage than the pet's attack. Yes, they're limited in number, but they're great utility and battlefield control. Its a hard call at this point, since we're talking limited resource and situational abilities. If we assume Hunter Marks, then we've the Beastmaster coming ahead with a extra 5 damage a turn compared to a Hunter. However, at this point, we're running into more and more creatures with resistances to mundane attacks, so that lowers the companion's output significantly. As well, we won't be using Hunter's Marks constantly, as we have access to Ranger-y AoEs, further skewing the data.
Swift Quiver blows the pet out of the water at level 17.
As for healing? I think that woiuld be fine, personally.