You can't say Ti or Te is more objective, because each one ties with a different functions.
In some ways, I am more objective than my ISTJ father, because he's swayed by tertiary Fi values; whereas mine are predominant but balanced out by Se, which is a completely objective function. Just because he uses higher ranking Te than I do doesn't make him better at observing fact.
Speaking as someone who thought she used Ti for a long long time, Ti and Fi are fundamentally similar. They are introverted judging functions; and while it is easy to normally tell them apart in an individual, it's not always so easy to tell them apart in yourself. In fact, I refused to think I used Fi for a long time because I was so attached to the word 'objective'--until individuals in reality pointed it out to me. And then I understood it.
Both Ti and Fi like asking questions; Fi reacts, though, immediately to the answers--while Ti is altogether less impacted by those answers. Fi may continue to ask questions until it realizes how it feels about a certain topic, whereas Ti will continue pursue in order to conceptualize the information into a broader framework--pass it on to Se/Ne to figure out what to do with that information--and ultimately Fe to express its thoughts.
As far as Te goes it likes empirical data and evidence. But if the first function is Si, then it is going to be biased. And if the first function is Ni, it is going to use whatever information it gathers to focus on a goal--which can be unhealthy. The same can be said if the functions are swapped. Just because a function sees reality objectively doesn't mean that the person using the function is being entirely objective because of the other functions that come into play.
All in all that's why I don't like the use of the words objective and subjective functions; external and internal are much better terms, especially when learning how this stuff works.