That sounds about right to me. The British at the time had a very large empire and India was a vital part of it. It also gave them sea ports for trade and to keep warships about in case of trouble.
There were two main reasons for the interest of British in India. Firstly the highly profitable trade with India. Secondly, the strategic importance of India which meant they could keep an eye on their possessions in Middle East and Far East? Correct?
India was important to the spice trade. There was also trade to be made in fabrics and gemstones and precious metals. Curiously enough, India was not THAT strategically important. British possessions with better facilities for warships existed all around the Indian Ocean. Deep water ports are rare in India (a problem that the Indian Navy still has to cope with). The ability to garrison India with fighting men was important however. Even at the height of British involvement, the ex-patriot army never exceeded 160,000 men. India was also important as a source of manpower, therefore. This is a very difficult question to answer.
I think what you've said is more or less spot on. India rapidly became a very precious trade asset, a fact that the British recognised very quickly and as such, decided to tighten it's grip by empirical rule, which also allowed them better access to other areas of the east.
In these times, as now, trade meant money, which meant power, and they became able to control areas where wealth and power were not so readily available- like the subcontinent.
The company was able to effectively rule parts of India with it's own private armies- they were not even reliant on support from the crown, although this changed towards the late 1800s, when Queen Victoria's England took direct control of British India.
Main trade commodities were as follows: Tea, cotton, opium and silk.
Meanwhile, the geographical proximity of India gave Britain a presence in the area, as events in the near and far east developed- the discovery of oil in Persia in 1908, for example.
Remember that the world of this time seemed bigger. Travel took longer and communication was more difficult. In order to be at the forefront of international events, countries needed a foothold in the area.