Is Sun brighter than what we actually see?

Note that the intensity of observable radiation coming out from stars would now mostly depend on the outermost layer.

Because the material in the star is opaque, it completely depends on the outermost layer. Of course the properties of that layer (such as its temperature) are driven by the energy coming from the interior.

So, wouldn't it be inappropriate to consider stars as Black bodies while determining their temperature and other properties?

It is appropriate for an object that has a spectrum that closely matches a blackbody spectrum. However the only property that describes is the temperature of the visible layer. It doesn't imply anything about the interior and processes that produce and distribute energy. You shouldn't read blackbody and think that means that the (invisible) interior is simple or in some way similar to the exterior.

The sun's interior is much hotter than the exterior (around 15 million Kelvin in the core, compared to the 6000 Kelvin or so at the photosphere). Because it is not visible, I would hesitate to call it "brighter". But you could consider it that way.

Yes, you are looking at the outer layer

All the heat and light from the Sun is emitted by that outermost layer, which is essentially opaque to the processes going on inside it. The solar spectrum is a very good fit for a black body, and thus a black body temperature calculation applies to the surface of the Sun. It's not that hot, at around 5700 K.

Yes, inside it gets hotter. A lot hotter

5700K is not hot enough for fusion, unfortunately. As you correctly guessed, most of the light generated in the core stays there. At the core, the temperature is millions of Kelvin, and more than a hundred times the density of water as a plasma.

Remember the Sun is a big gas/plasma ball

The size of the Sun is governed by the balance of forces; outward pressure from the plasma which wants to expand vs inward pull of gravity. If there was more energy generated, that would tend to make the ball expand - there's no 'surface' to constrain it. So the surface and its properties are entirely a product of the energy being generated within the ball, and in fact largely at the core.

What escapes is not a small fraction of the energy being generated

The Sun is in a (broadly) stable state - it's not getting bigger or hotter. And the energy released by fusing nuclei doesn't have any other place to go; it's not going to go back into the heavier nuclei and split them up. So if the Sun is stable, that energy must all be flowing out in one way or another.

We know, then, that the total energy being emitted by the Sun's surface is equal to the amount of energy being released by nuclear interactions at the Sun's core.

We aren't saying the Sun is just a hot blob

By modelling the Sun as a black body, all we are saying is that we can see its spectrum matches that of a generically hot thing at a particular temperature, and from there (and some assumptions about it being basically the same in all directions) we can work out the total power output of the Sun. We haven't had to make any assumptions about what's going on inside the Sun, only observe what reaches us and infer what must be coming out overall.

With this knowledge we can start to infer what must be happening inside

We know some things; how much power the Sun is producing, how big it is, the proportions of atomic species at the surface (from the details of the spectrum), its overall mass, the laws of gravity and the behaviours of plasmas. From that we have to try to model what must be happening inside.

We have inferred, for example, what the dominant nuclear processes inside the Sun must be; we know the temperatures and pressures are not high enough for some cycles, but high enough for others.