Mixing Grounded and Grounding
If this panel serves as the main disconnect, and the grounding and neutral bus bars are bonded. Then yes, you can terminate both grounded (neutral) and grounding conductors at the same bus bar.
Notice that the bus bars are bonded using a bonding strap.
Multiple Conductor Terminals
If the bus bar is able to accept multiple conductors under some or all of the terminal screws, it should be labeled as such either on the bar itself or in the panel documentation. Terminating multiple conductors under a screw that is not rated for it, can cause bad connections, excess heating, damage to the conductors, etc.
Additional Bus Bars
Some panels are configured to allow additional bus bars to be installed. If your panel is one of them, you could install an additional bus bar and terminate extra conductors there. Other panels may accommodate extended (longer/larger) bus bars, that can be installed in place of the existing bus bars.
Ground and neutral are not parallel neutrals. I know it looks that way because they're bonded in the main panel. But shift into a different way of thinking about the purposes of the 2 wires. Think of the ground solely as a safety shield.
Let's try a few pairs of examples. The first is Code and the second bonds at the sub-panel also. The orange glow is on things which are "hot".
Seems awesome right? Poor old Code Man is in the dark. His power tried to return via neutral, and neutral is broke, so the power failed. Rogue Man is one happy guy and his life isn't disrupted. Ground is working great as a "backup neutral". He doesn't even know he has a problem!
Of course, ground is a thinner wire, so it might overheat, but so what? Or, what if both ground and neutral were cut?
Code Man is still in the dark and he's still gotta fix those wires. Rogue Man is dead.
In Code Man's installation, the hot went through the bulb, looking for neutral. It didn't find it, so it pulled the neutral up to 120V ( not enough power for useful work, but plenty to shock). It did the same for Rogue Man, but since he tied neutral to ground in the sub-panel, ground is now also 120V, including the service panel cover and the switch plate cover screws.
Suppose the sub-panel has its own ground rod. That doesn't help much. Earth tends to have high resistance, so the cover screws might be 103V instead of 120V.
I have the good fortune of working in EMT conduit in a steel building, which naturally forces the entire conduit system to ground. Ground is never part of the circuit in any way whatsoever. So I get to see it as intended, as a protective "shroud" around all things electrical.
Ground isn't quite yet a perfect envelope. It is in new work, but we still have a lot of old wiring out there that is not practical to outlaw entirely - such as NEMA 10 and switch-loop smart switches which poach ground as a neutral.
Why bond neutral at all?
That's a GREAT question. Not bonding ground would give you an isolated system. And that makes a lot of sense in some ways, like solving some of the problems you see above. But it has other disadvantages. I go into depth about that here.
Best Answer
I think you may be over-thinking this. You can make a connection on either bus without reaching across an ungrounded bus in that panel.
The ground and neutral bars are so located as to give you free access to them without have to reach around or near an energized bus. You could stand to the right or left to make your connections without reaching across the phase bus.
In a larger piece of switchgear, the phase conductor bus should be located further in and the ground and neutral bars closer to the access panel not the other way around.
I think that is the intent of the code.