c = 1 in natural units. And in natural units, it is natural to consider c to be dimensionless. It is legitimate to assign Time and Length the same dimensions, so that speeds become dimensionless numbers. In principle, you could do that in any unit system, but the fundamental physics equations strongly suggest that this should be done in natural units.
So, we can say that c = 1 and that 1 is not a large number at all.
The reason why c is large in SI units is because of the way we decided to define the meter and second. Note that it doesn't make sense to consder a dimensionful number to be large or small. So, if you say that c is large in SI units, what you mean is that:
c* second/meter is large
Now, if we evaluate this in natural units in which c = 1 (and dimensionless), this tells you that the second is huge compared to the meter.
So, relative to a consistent definition of the unit of time relative to the unit of spatial distance, we have decided to use inconsistent units for time and spatial distances. For historic reasons we decided to define units so that the older definitions would still be approximately valid. And a long time ago the smallest units for lengths and time intervals that were used a lot in practice were the smallest quantities that were still relevant for humans.
This means that the reason why c is large in SI units is because we are very slow. We can only perceive changes that happen on extremely long time scales compared to our size. If things happen too fast we perceive that as in instant change, we don't see that the change in fact happened gradually.