MiraMels
Explorer
If you buy a warhorse or a hunting dog, then I would expect that they're going to be capable of combat, and if you give them their head, they will do more than run off.
Will they be perfect tactical companions? No. But they're going to attack, and it won't cost you an action to make them do so.
The paladin summoned warhorse is a step up from that - it's loyal, obedient, extra intelligent, share spells with you and are telepathically linked to you. So why would it they be incapable of independent action? Again there are disadvantages compared with the beastmaster: namely that they have fixed hit points, defenses and attacks that will rapidly become outclassed.
A warhorse is just a horse that has been trained to follow your orders under stress, and not panic at the sight of blood or the sounds of battle. It's not trained to like, discern friend from foe, and kick monsters at your command. If you dismount, they'll try to stick around, and defend themselves if they can't run away, but like, your best bet for keeping your warhorse in the fight is to stay on the horse.
Mastiffs are in there because Small PCs exist, but if they aren't serving as mounts, I'd expect them to be able to serve as watchdogs (as in, make noise when they see someone they don't recognize, and attempt to threaten and waylay them), or help you track down and flush out game (but not actually kill them, that's your job as a hunter), or maybe you can get a group of them together, point them at a bunch of men across a battlefield, and set them loose. But they are severely limited in the kinds of creatures and situations where they'd be willing and able to fight. You can't expect an uncontrolled mastiff to do anything but run from a hippogriff, or a flameskull, or a hill giant. There are foes and situations that are simply above their training and capabilities.
(An important thing to note, here, is that the Chapter 5 mounts are under the DM's control whenever you are not sitting on them. The player is not moving the creature or declaring actions on their behalf. You can issue commands to them, within their training and capabilities, but even that might require a Wisdom (animal handling) check. And, besides, giving a command of that nature in combat would be an Improvised Action; it would use your character's action to verbally command your mastiff to attack, just like the Beast Master.)
But if you want a canine companion who is capable and steadfast in the face of any danger, who is loyal enough to follow you into any of the Nine Hells, and is exceptional enough to be a damage-dealing combatant, no matter the foe? That's the Beast Master's animal companion. (or it's supposed to be anyway. Your mileage will vary, and you've made your opinion of the beast master clear. If you are disappointed with that offering, then I'd invite you to modify it as you see fit, until you are satisfied that it can fulfill its intended role. The Chapter 5 mounts are not intended to serve as damage-dealing combatants.)
As for the paladin, I am quite confident that the intention of the spell was to grant you an exceptional mount. One that remains under your command even when you are not sitting astride it. It makes sense that the Paladin would be the best class at mounted combat but I cannot imagine that a second level spell of theirs was ever intended to replicate an entire subclass. (Yes, even if the Beast Master's Companion will eventually be better.)