According to Halacha and the Torah, are we still obligated to do capital punishment like stoning people for sins like breaking Shabbat? Is this rule kind of swept under the rug or is there a reason why we don't do this? Rabbi Elazar ben Azarya states: “[More than] one person in 70 years [would be denoted a murderous court]." can someone please explain this?
Excellent question. The answer to your question can be found in Sanhedrin 41a
ארבעים שנה קודם חורבן הבית גלתה סנהדרי וישבה לה בחנות ואמר רבי יצחק בר אבודימי לומר שלא דנו דיני קנסות דיני קנסות סלקא דעתך אלא שלא דנו דיני נפשות
Rashi explains that the capital punishment cannot be applied only if Sanhedrin is in the Lishkat Hagazis (which is located in the Beis Hamikdash) as it says (Deut 17:8)
וְקַמְתָּ וְעָלִיתָ אֶל הַמָּקוֹם אֲשֶׁר יִבְחַר יְדֹוָד אֱלֹהֶיךָ בּוֹ
from this we infer that the place of the Sanhedrin allows them to apply the capital punishment. if they are not in their proper place they cannot apply Dinei Nefashos. That is why when there were too many murderers in the land, they moved from the Lishka so they wouldn't have to kill too much (the reason is unclear, perhaps they were afraid of the Roman empire, or because they didn't want to kill so many people as we will see later). In any case, it is clear that Dinei Nefashos cannot be applied only through a proper Sanhedrin in its proper place!
Then there is the other problem with Semicha. The ruling body of the Sanhedrin needs to be ordained by others who were themselves ordained.... til Moshe Rabeinu (see rambam hilchos sanhedrin 4:1). Semicha is lost to us since the times of the Chachmei hatalmud when it fell into disuse, and since then was never recovered.
As for your other question why Chazal were against killing too many murderers? from the Mishna in Makkot (7a) it is evident that Chazal had an aversion to the death penalty and thus tried to limit it as much as possible. R Akiva was so reluctant to apply the death penalty that he declares that had he been in the Sanhedrin no one would've been killed (by finding loopholes as described in the Gemara). By adding lots of Dinim and requirements (see Sanhedrin 40b) as to when the Sanhedrin is allowed to kill they effectively limited the Dinei Nefashos. But as i already said that is not "the" reason we don't do Dinei Nefasos anymore. It is because we don't have Sanhedrin!