In the 1950s, many US rockets were capable of reaching the lower limit of LEO altitude. Despite this, no one thought of making the launch vehicle / its payload orbit around earth, even for one revolution till the Russians did it on 4th Oct. 1957. Why so? Did the rockets then, were mere like advanced fire crackers, reaching very high altitude, and falling back on earth? Or was it because the technology to make something "orbit" around earth was snot developed then?
It's a matter of velocity. These rockets weren't capable of reaching Earth orbital velocity which is about 4.8 mi/s (7.7 km/s). Only at this velocity, when flying about parallel to the surface, the orbital perigee would no longer be within the Earth so that the craft would fall around the entire planet instead of falling onto the surface.
V2 rockets for instance could only fly a parabolic arc and fall back to Earth again. The Soviet R-7 that launched Sputnik 1 was the first rocket that could reach orbital velocity, and in America the Vanguard and Redstone rockets.
Despite this, no one thought of making the launch vehicle / its payload orbit around earth, even for one revolution till the Russians did it on 4th Oct. 1957. Why so?
This is not the case.
Both the Soviet Union and the United States were extremely interested in the capabilities that Nazi Germany had developed during World War II. Both attempted to capture (and succeeded in capturing) as much of the German rocket technology and technologists as they could. The Soviet Union did a better job at capturing the technology while the US did a better job at capturing the technologists.
The rationale for this interest was that both sides knew that the short range capabilities of Germany's V2 rockets represented a good start at a military capability that would be far more potent than the ability to bomb London from rockets launched from Holland. To be able to drop a rocket-based bomb on a city in Russia from a rocket launched from the United States (or to drop a bomb on a city in the United States launched from the Soviet Union) required significantly more thrust than the limited thrust from a V2. What Nazi Germany had accomplished was a good start, but only a good start.
Being able to drop a rocket-launched bomb on a city that is halfway around the world (as opposed to being able to drop a rocket-launched bomb on London from rockets launched from Holland) requires (near) orbital capabilities. The early stages of the space race between the (former) Soviet Union and the USA were driven by purely militaristic concerns. It was only when the two nations developed the militaristic ability to send objects into space that peaceful uses of space became apparent.
Getting to space is easy. Staying in space is hard. Staying in space is not about going up high. It is about going sideways really fast.
From Navi Mumbai to Hyderabad is 600 km, space is only 100 km away. So, you are about 6× closer to space and about 200–300 km closer to the ISS than to Hyderabad, according to your profile. With a high-performance car, you can drive 100 km in 30 minutes.
A small sounding rocket can go to space. Some experimental airplanes could go to space by accelerating and then pulling up, trading speed for altitude.
The problem is staying there. It's not like you reach space, and gravity suddenly turns off. In fact, at the altitude of the ISS, gravity is still about 90% of what it is on the Earth's surface, plus even at 400 km, there is still a tiny bit of atmospheric drag that slows down the ISS so that it needs to be re-boosted regularly.
Here's how orbiting works: imagine you have ball in your hand, and you drop it. It will fall straight down and land directly underneath the point where you dropped it. Now imagine you throw the ball just a little bit: it will still fall down at exactly the same rate as the ball you simply dropped, but it will also move forward at the speed that you threw it at.
Note that we are ignoring air friction here. The reason the ball drops down and stops flying is not because it is slowed down by air friction. It is because it is being pulled down by gravity. You can still play golf on the moon, for example, and the ball will not just go off into space.
Okay, now imagine you throw the ball harder: it will fly further until it hits the ground, and if you look at it from the side, the curve the ball follows will look shallower. And we can imagine shooting the ball out of a cannon, which will make it fly even further until it hits the ground.
All that "orbit" is, is that you throw the ball so fast that it falls down at the same rate as the curve of the Earth makes the ground "fall away". In other words, we want to make the curve of the ball match the curve of the Earth. So, you continue falling and falling falling, and the Earth is curving and curving and curving, and as a result you never stop falling and always "miss" the Earth.
It turns out that the speed needed for that, which we call Orbital Speed is really high: if there were no air (and no mountains that we could crash into), we could orbit just a few millimeters above the Earth's surface at about 7.9 km∕s = 28 440 km∕h or over 140 times faster than our 200 km∕h car from above. Orbital speed at LEO is still 6.9–7.8 km∕s = 24 840–28 080 km∕h.
Just for comparison: if we were to take off at this speed, it would take us less than one minute to reach the ISS. But actually, most rockets are not even out of the atmosphere yet after one minute.
As you wrote:
In the 1950s, many US rockets were capable of reaching the lower limit of LEO altitude.
Yes, exactly. They were able to reach this altitude, barely. Then they were out of fuel and fell back to Earth.
In order to orbit, after reaching there, they would have had to turn sideways 90° and then accelerate from 0(!!!) to orbital speed, at least enough for an eccentric orbit that keeps them out of the thicker part of the atmosphere most of the time. (In reality, it is not very efficient to go straight up, then turn 90° and go straight sideways, instead, you gradually turn shortly after leaving the launch pad.)