Eastern liturgy is not the only issue when speaking on this topic. Addressing this a simply Eastern Rites is to look at this from an wholly Latin perspective, greatly distorting the matter. What is being discussed is eastern ritual CHURCHES, not merely different Rites. The Latin Church sui juris is unified in jurisdiction under the Papacy and has (today) very few rites, namely Roman, Ambrosian, Mozarabic, as well as some local usages. The Eastern Churches are 23 distinct Churches from the Latin Church which total collectively (all 24 Churches) comprise the communion that is the one, holy, catholic, and apostolic Church. These 23 Eastern Churches have a wide variety of liturgical Rites. Some are Byzantine, like the Ruthenian, Ukrainian, and Melkite Churches. Some are Syriac, like the Chaldeans, Maronites, and Syro-Malabar Churches. And still others are Alexandrian in their liturgy, like the Coptic and Ethiopian Churches. Latin Catholics can freely attend the liturgies of the Eastern Catholic Churches, but formally joining them is an altogether different issue since that is moving from one Church to another Church. It is easier today to switch Churches, but once had to be done through a formal process at the level of the Roman curia.
I am wholly comfortable in the Byzantine Rite in its Slavic recension. In fact, I live much closer to both Ruthenian and Ukrainian parishes than to where I attend the Tridentine Mass. However, I am a Latin Catholic and will always call the Tridentine Mass home.