What is the purpose of 了 in these sentences: 1:找到你了。(If it is completion marker, why doesn't it come right after the verb?) 2:我最喜欢看篮球比赛了.
This dictionary definition addresses the use of 了:
E.g. 小王来信了 / 都四月份了，天还这么冷 / 天快亮了 / 要是不走就见到他了。
Basically, it says this use of 了 is placed at the end of a sentence, introducing a new circumstance or change. The new circumstance can be something already happened, or will happen, or a hypothesis.
So, 找到你了 // finding you is a new circumstance which already happened.
我找到你 - "I find you". This sentence lacks clarity on whether the act (find) has happened in the past, now, or is ongoing.
我找到你了 - "I find you now" or "I found you". These two indicate the act is either completed now (the former) or was completed in the past (the latter).
我最喜欢看篮球比赛 = 我最喜欢看篮球比赛了 - As this sentence is a general statement that does not involve tense (no action to complete), so with or without 了 makes no difference, only up to the style of speaking one's accustomed to.
了 means 'at present'
So 找到你了 means, as present, I have found you and 我想家了 means, at present, I miss home. When you add 了 after the verb the meaning changes 我去找到了你 means I went to find you, and 我想了家 means I thought about home.
Interestingly you can also say 我找到你啦 where 啦 sounds like 了 but 啦 indicates exclamation. An example is 她们赢啦！ （they won!）Another example is 太贵啦 which you say when you bargain, it means 'this is too expensive' but you can also say 太贵了，我没买 which is not as emotional/exclaimation-al, and you are just trying to say 'too expensive, I didn't buy it.
了 is a particle with no actual meanings, but there are three kinds of particles: 状态助词, 语气助词 and 结构助词. 了 is both a 状态助词 and a 语气助词.
When it is used as a 状态助词, it shows the thing the speaker talks about is finished or started but have not ended, and should be after the verb or verb phrase; when it is used as a 语气助词, it is just a sign of statement and must be at the end of the sentence.
了 here is both a 语气助词 and a 状态助词, for it is at the end of the sentence and 找到你 is a verb phrase.
了 here is obviously a 语气助词.