My head-canon is that there's nothing in the technology actually requiring this, it's just how they built it. Just like there's nothing preventing your car from having its gas tank in the roof, but nobody builds them that way.
At the time when Omni technology was created, PPCs, Gauss Rifles, and Autocannons were expected to be the "primary guns" of a mech. Given that the Clans don't really engage in melee combat that much, keeping the hands in place while carrying heavy weapons wasn't really considered that important. Basically the designers decided this was an easy shortcut, and they took it. The lower arm and hand assembly would be considered an easy detachment point. Just unplug the lower arm from the elbow connector, and it removes easily. Then you can stick your weapon pod in its place with a minimum amount of fuss. The advantages you get from being able to flip arms would outweigh the inability to punch, at least in the Clan mindset. And if you really need to keep your hand actuator for some reason, you can just use a Large Pulse instead of that PPC.
Now, there's nothing requiring an Omnimech to be built that way. You could theoretically design it where you could attach a PPC underneath the arm assembly. But they didn't. Maybe there are good reasons not to do that, ones that don't show up in game stats. I've always felt that designing a mech in-universe should be harder than it is for us scratching it down on paper. Just because something would be game legal doesn't mean that it would always work in-universe. It's not just a matter of counting available critical slots and tonnage.
As far as lasers and machine guns go, I presume they aren't bulky enough to require the entire arm assembly to be removed. It's easy enough to squeeze them in somewhere without jacking with the external housing too much, or throwing off the balance of the machine. Again, these are things that would be clearly understood by a Clan tech, but we the readers just don't know enough about how it works to know why they did it this way. If somebody is applying the "no hand" rule to AP Gauss weapons, then I think that's just carrying the general rule too far and it's a dumb decision by a game company employee.
With LRMs and larger SRMs, I think the assumption is that even if an LRM-20 is mounted in the "arm", it's probably much more of a shoulder mount. It's something that ends up looking like a MadCat's missile launcher instead of the hand mounted one that the Salamander carries. In game terms it's in the arm, but visually it isn't going into the lower arm area (even if that's where the crit spaces are). Some versions of the Crusader are drawn this way.
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While there's nothing requiring Omnimechs to look a certain way, I think the fact that the early Omnis all had a certain style supports the idea that a lot of their parts are made to be interchangeable. A Loki arm versus a Thor arm versus a MadCat arm, they're all the same basic exterior housing. The big improvement with Omni designs would not be in the mech, it would be in the supply chain. All your PPCs are made with the same type of connector. All of them are made to fit certain dimensions. An Inner Sphere PPC might look like a long, thin barrel (i.e., on the Warhammer), or be a short, fat double-barrel (like on the Battlemaster), or the underslung version on the Panther. But the Clan versions (at least post-Omni) are all going to be very similar, because they're all plug-and-play.
Once that standard is established, it doesn't make any sense to stray from those norms. Yes, you could develop your own Omni that could carry an underslung Gauss Rifle and still have a hand. The Nightstar carries one in the arm and has a hand, after all. But you're basically redesigning a major part of the system from scratch. And do you really gain that much from it?
Now, later Omnimech art kind of throws the whole modular appearance thing in the toilet. Actually I think that's where most of those designs belong (completely horrible mech art in later TROs). But at least as far as the original Omni designs, they all look pretty interchangeable.
