What is the importance of measuring/calculating antenna G/T? As I understand, it tells us something about the receiving performance of the antenna, correct? A higher G/T ratio means the antenna has better sensitivity to weak signals, resulting in improved overall receiving performance. Thus, G/T is critical in designing and evaluating antennas, especially for communication with distant or low-power signal sources, such as satellites. Can the manufacturer's G/T value be trusted all the time?

You ask "Can the manufacturer's G/T value be trusted all the time?"

I would say definitely YES, assuming you have placed a contract for an antenna plus low noise amplifier and the manufacturer has installed, adjusted and tested the antenna plus low noise amplifier. It is essential that you implement a contract acceptance test to show that the specified G/T is achieved. So the specified G/T will be met. Note also that achieving the specified antenna patterns and polarisation discrimination is similarly important. If it is a transmit antenna it must pass all testing to be allowed to operate by the satellite operator. 

If you contract out the whole task you don't have to worry, but the cost will be higher. 

You can save money if you buy the antenna (possibly second-hand or as parts) and assemble and adjust it yourself and choose your own low noise amplifier. It is up to you to make a design to meet whatever G/T you want to achieve. This means getting the gain high enough and attaching a sufficiently low noise temperature amplifier with a minimal loss waveguide. It is then up to you to get the antenna working well and carry out all the tests specified by satellite operator. You take the responsibility.

Getting the sidelobe patterns and polarisation discrimination right has a lot to do with the antenna and feed manufacturers original design plus the skill of the assembler/installer in terms of panel alignments and feed positioning.

See these links:

System G/T calculator

Examples of antenna noise temperature at different elevation angles

Subreflector alignment

Thank you