Trying to assess this with RAW is a misstake.
Any ruleset needs to be tempered by adjudication. Apply common sense. If X doesn't make sense to the GM then they will say no. Even if X is RAW and you can show it to be.
The common-sense of the player should inform them on whether to even try to do a given something, the GM at that point will tell the player if that has a chance to work and how to resolve it if necessary.
When rules seem to say something that does not make sense, ask the GM.
Logically, the existence of these saddle-items imply that an animal cannot carry any loose objects unless it has those saddle-items. Note how no rule prohibits an animal from holding on to items like some other creature, nor is there a rule to stop it from wearing a backpack (or multiple).
What the rules do say is that all creatures have item slots based on their body type (And that a maximum of one magic item in each such slot functions normally), nothing is said about slots for non-magic items. There is no slot or such defined for saddlebags. Can an animal even hold a saddle? Only the existence and description of the items themselves imply that animals can wear them.
Note also how nothing tells you how to strap something on to a creature using rope or glue. And nothing limits how many saddles you can stack on top of the same animal.
Is it possible to put both a pack saddle and a military saddle on the same animal? RAW? Yes, but... no sane GM will let that fly. If they both are saddles then there is no sensible way to put them on the same horse. Rules wont tell you that a horse is limited to a single saddle at a time.
Is it possible to ride an animal "bareback" if it already has a pack saddle?
RAW? Yes, but...
Though the pack saddle doesn't say you can't ride the animal at all, it does say the saddle cannot accommodate a rider. And any GM who knows what a Pack Saddle looks like and who cares about a consistent world is going to say no, you cannot ride a horse with a Pack Saddle.
The common sense that tells us we cannot stack multiple saddles and that the saddle occupies the space where a rider would sit leads us to conclude that you cannot ride the animal, at least not normally.
Much of these types of questions where the rules suggest something bizarre can be answered by simply taking a step back from the rules and recalling the purpose that rules serve in any TTRPG (At least those with one sole GM). The rules purpose is not (primarily) to let the players know what they can or cant do! But rather they are guidelines to the GM on how they can adjudicate things. And whatever the rules say, the GM decides if it even applies, and they can rule in direct opposition to these rules if they choose.
Sharing the rules with the players and having discussions on RAW is not the default position for such an RPG. Sharing the rules with the players is just a convenience that helps set expectations on how the game works and helps things run smoothly when the players handle their own characters mechanically.
This is true for every TTRPG, its just that most games and "the meta" have shifted so far into the territory that the rules are the domain of a publisher and that RAW is sacred, leaving the GM to adjudicates whenever the rules are contradictory and when something does not make sense the player turns to websites to find other that echo what the rules say under a literal "RAW" reading.
That's a lot of words to say: Talk to your GM about how saddles work in his game. If you are the GM, how do you want saddles to work in your game?
In your particular case, you are dealing with a Mastodon. That's quite a lot bigger than a horse and I can imagine most GMs would let you ride it and let it carry stuff and let it fight all att the same time, but it would probably not be using regular, horse-sized saddles. I do not know that exotic saddles apply to such a large animal.