Do we really have 3 equal branches of government?

Associate Justice Renato Corona is now the new Supreme Court Chief Justice, this is the news that's been selling hotcake on our local media nowadays. A known ally and close contact of President Gloria Macapagal-Arroyo, his appointment has been the current target of criticsms from different groups and people. From the point of Mrs. Arroyo's 90 day ban in appointments, to his just being so-close to the Lady President. This prompted me to think and analyze, does our government really has 3 co-equal independent branches?

Recalling from college in our political science subject it has been taught that our government is composed of 3 co-equal independent branches, the Executive, headed by the President, the Legislative headed by the Senate President and the House Speaker, and the judiciary headed by the Supreme Court Chief Justice. As to what is indicated in the constitution these 3 branches were to act separately and or independently with each other. They have their autonomy as to how to run their respective branches, or in other words each branches should have no influence with one another's own processes and decisions. The president cannot dictate the legislative what to do, as well as the judiciary and vice versa. Although it is common knowledge that the legislative serves in the pleasure and anointment of the executive, might as well it is true with the judiciary.

But to take a closer analysis, we can give benefit of the doubt to the legislative because their leaders (Senate President and House Speaker) are elected by their own members meaning all the senators and congressmen/women. But if we look at the judiciary, all the justices and the chief justice are appointed by the president, isn't logical to think that the judiciary is under the executive branch and not a separate co-equal branch. Could a separate entity have a leader appointed by the other branch and claim its independence? Maybe I am missing some points here, but on a plain view analysis, its hard to digest how can they be independent in all the decisions they make if you are appointed by the executive branch. To put this in a more simpler example, we have 3 neighboring houses, the 1st and 2nd house has its leaders appointed by members of its houses, but the 3rd one has a leader appointed on by the 1st house's leader, isn't not safe to say that the 3rd house is not independent  but rather is under the 1st house because the leader is delegated by the it?

I think this is the loophole of our government's setup, we can never have a full, honest and sincere independent judiciary for as long as its head and members are appointed by the executive branch. The president will always have the perceived influence on judiciary's actions because he/she is the one responsible for their positions. Its like a master/servant relationship, how can you not give favor to the one who have given you the power, or more likely an 'amo' - house manager relationship, how can you ignore your 'amo', who have given you the power to manage/lead his/her other houses. This is what our lawmakers should take look at and put into consideration. If it is framed in the constitution, might consider reviewing those provisions and start creating amendments that will put the judiciary's independence in a more reasonable and more logical look on a plain citizen's view.

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"A government of influence is a government of patronage"...