In Tora
Thanks to DoubleAA, I understand a little better the OP.
Baal keri status originally concern uncleanness and cleanness laws and not prayer or learning Torah; see Mishna Kelim 1, 1 (semen makes unclean the person that touches it and the person from witch it originates). The verses cited above in the OP are part of those laws.
A large number of verses matters laws of uncleanness-cleanness. There are performative statements. You can see also that "Sifrey Mitsvot" include him in the 613. But it must be carefully examined. I will quote one excerpts from verses and from Sefer Hachinuch: A verse (numbers 19, 19) says:
And the clean person shall sprinkle upon the unclean on the third day, and on the seventh day; and on the seventh day he shall purify him; and he shall wash his clothes, and bathe himself in water, and shall be clean at even
Chinuch 399 שנצטוינו בדיני מי נדה, כלומר, בדיני מי הזיה, דהינו מים חיים מערבים באפר פרה, שמזין בהם על הטמאים
We were ordained to follow laws...,
There is a machloket between Rambam and Ramban if we should count laws of cleanness-uncleanness among the 613. So the majority of this laws says: in order to become clean, he shall bathe all hisflesh in water...
Nowedays, is Baal keri subject to specific laws?
There is a Machloket between Rif and Rambam.
This topic is covered in massechet Brachot.
mishna Chapter 3,4.(in Gemara folio 20B)
A BA'AL KERI (a man after having seminal emission) SAYS THE WORDS [OF THE SHEMA MENTALLY WITHOUT SAYING A BLESSING EITHER BEFORE OR AFTER AT MEALS HE SAYS THE GRACE AFTER, BUT NOT THE GRACE BEFORE. R`JUDAH SAYS: HE SAYS THE GRACE BOTH BEFORE AND AFTER (Soncino)
I will quote the Bartenura on the Mishna concerning baal keri. as an abstract of Gemara and rambam, and then I will try to explain through reading a few lines of Gemara.
Bartenura:
עֶזְרָא תִּקֵּן שֶׁלֹּא יִקְרָא בַּעַל קֶרִי בַּתּוֹרָה בֵּין שֶׁרָאָה קְרִי לְאוֹנְסוֹ בֵּין לִרְצוֹנוֹ עַד שֶׁיִּטְבֹּל. וְלֹא מִפְּנֵי טֻמְאָה וְטָהֳרָה שֶׁאֵין דִּבְרֵי תוֹרָה מְקַבְּלִין טֻמְאָה אֶלָּא כְּדֵי שֶׁלֹּא יִהְיוּ תַּלְמִידֵי חֲכָמִים מְצוּיִין אֵצֶל נְשׁוֹתֵיהֶן כְּתַרְנְגוֹלִים ... וּכְבָר נִפְסְקָה הַהֲלָכָה דְּבִטְּלוּהָ לִטְבִילוּתָא, וּבַעֲלֵי קֶרְיִין קוֹרִין קְרִיאַת שְׁמַע כְּדַרְכָּן וְעוֹסְקִין בַּתּוֹרָה וּמִתְפַּלְּלִין וּמְבָרְכִין כָּל הַבְּרָכוֹת וְאֵין מְעַרְעֵר בַּדָּבָר
Ezra ordinate that a man after having seminal emission do not read Torah, no matter if this emission was adventitious or deliberate, until he immerse in Mikve. The reason for this is not uncleanness, since words of Torah are not susceptible of uncleanness, but the aim is to avoid that scholar might always be with their wives like cocks.
Ezra's ordinance about men with a Status of baal keri
Torah doesn't need cleanness: See Gemara Brachot 22a
On the basis of this they laid down that sufferers from (gonorrhoea) [urethral discharge, gleet], lepers, and those who had intercourse with niddoth are permitted to read the Torah, the Prophets and the Hagiographa, and to study the Mishnah, [Midrash] the Talmud, halachoth and haggadoth, but {despite uncleanness they may be in fear and trembling (according to Rashi)} a ba'al keri is forbidden {his uncleanness is a consequence of lightness and pretentiousness}.
The ordinance of Ezra
Ezra ordinates a law concerning this specific uncleanness. A man after having a seminal emission is not allowed to read Torah until he clean himself in a Mikve.
The ordinance is not a "pure novelty", see Gemara 21B about matan tora.
But has not R`Joshua B`Levi said: How do we know that a ba'al keri is forbidden to study the Torah? Because it says, Make them known unto thy children and thy children's children, and immediately afterwards, The day that thou stoodest [before the Lord thy God in Horeb], implying that just as on that occasion those who had a seminal issue were forbidden, so here too those who have a seminal issue are forbidden?
We learn from matan Torah a link between Torah and cleanness (Tum'at keri, 'uncleanness from semen').
Bet Din after the generation of Ezra extended the ordinance to prayer too (Rambam tefila 4, 4; as shows the next mishna, "he was praying and he recalled that he was baal keri...")
The intention of the ordinance
Rabbi Bartenura (following Rambam (sefer Ahava hilchot Tefila chapter 4, halacha 4)) says that the purpose was in order that scholars might not always be with their wives like cocks.
We will see that this is an interpretation of the Gemara and not a citation of the Gemara. The sources are probably 3 passages of Gemara: 1.- the gemara (22A) cited bellow that contain the expression "with their wives like cocks (teacher who was teaching in a low voice, that avoided to spread widely, that it suffice to use 9 kabin), 2.- the Brayta that point discrepancy between Baal keri and the other uncleans (zav, boel niddah, etc), 3.- and the following Gemara in 22B:
Ezra ordained that a healthy man whose emission is voluntary must immerse in forty se'ahs
They built a new statement. The legislative intent was to prevent an excessive amount of marital relation among Scholars.
But this ordinance don't spread in all Israel, and most people failled to follow it, consequently it would annulled (Rambam Kriat Shema 4, 8)
They are now allowed to learn torah, pray and bless.
In this, we follow the words of the Rambam.
Gemara 22A
R' Nahman Bar Isaac said: It has become the custom to follow these three elders, … and R' Judah Ben Bathyra in the matter of words of Torah. (see above in Gemara : R'Judah B`Bathyra used to say: Words of Torah are not susceptible of uncleanness) … etc.… When Ze'iri came [from Palestine], he said: They have abolished the ritual (ablution) [immersion].
Then, the Gemara comment a Berayita that says
A ba'al keri on whom nine kabs of water have been thrown is clean
And they debate about the dissemination of this teaching:
The one who taught 'he whispered it' thought that the reason was in order that scholars might not always be with their wives like cocks.
The Rif wrote:
איכא מאן דאמר האי דאמרי בטלוה לטבילוטא בין לדברי תורה בין לתפילה ואיכא דאמרי הני מילי לדברי תורה אבל לתפילה צריך בטיבלה ולאו טבילה דווקא אלא רחיצה בט' קבין. וכתב רבינו האי גאון זצ"ל כיון דגמרא ליכא בהאי מילתא נקוט מנהג כל ישראל שכל בעלי קריין אף על פי שאין להם מים אין מתפללים עד שירחצו.
Is the ordinance entirely or partially cancelled?
The Rif concluded following Rav Hayi Gaon that for prayer the ordinance has not been annulled (see also Tosfot) and Baal Keri need at least 9 Kabim.
But Rabenu Yona and Rambam do not agree and the Shulchan Aruch follows the Rambam and wrote that the "Ezra's immersion" was completely cancelled S.A. O.C. 88
Magen Avrohom, cited in Mishna Brura calls those the continue the minhag as Anshey maasse (people with a great behavior). This minhag is laudable (note that it is different from the 'daily Tevila'. Daily tvila is not needed if a man is not Baal keri. On the contrary, in Gemara we see several evidences that prove that scholar were not going to Mikve if they were not baal keri). {In Gemara, different types of Baal keri are distinguished, immersion in mikve and 9 kabim water thrown, in Rishonim, dinim of Chatsitsa, Kelim and Sheuvim. see in poskim}