Is it okay if I start a sentence with the phrase "In a (or one) sense" instead of "On one hand" while comparing two things? If so, what about usage of "On the other hand" in starting the counterpoint?
You use "in a sense" with meanings. As in:
I say (this). In a sense, it is true. In the other is not.
In this particular case, IF there are ONLY two possible senses, you can use "on one hand".
But there can be more senses to a statement. In that case, "on one hand" is not suitable, unless you are willing to get to the part with: "On the fifth hand...".
Additionally, "On one hand" can be used with anything, not only meanings of statements.
On one hand, I want to enjoy the hot sun. On the other hand, I do not want the ice cream to melt, considering I do not have a freezer.
As you can see, "on one hand" is a lot more generic, but applicable when there are only two options to be compared.