But I think that when you say that, “the Chair of Moses is that it belonged to Moses alone”, you forget, or perhaps never noticed Jesus own words on the matter. I refer you to this from Matthew 23 vs 2-3:
1 Then Jesus spoke to the multitudes and to His disciples, 2 saying: "The scribes and the Pharisees sit in Moses’ seat. 3 Therefore whatever they tell you to observe, that observe and do, but do not do according to their works; for they say, and do not do.
Peace
James
It amazes me that Catholics use this verse to support the CC,

All you have to do is keep reading and you discover that “holding Mosses seat” was no constellation prize on the contrary, it meant absolutely NOTHING! Jesus goes on to blast them straight to hell. The 7 woes to those who hold the seat of Mosses. I do agree that there is a likeness in the catholic church and the Pharisees, or better said the seat of Peter and the seat of Mosses, I really cant tell the difference.
C/P from ESV study bible.
23:1-12 Warnings against the Teachers of the Law and the Pharisees. Jesus warns the crowd and his disciples not to follow the false leadership of the Pharisees (v v. 1-12), then directly pronounces woes upon those leaders for their deadly actions (see v v. 13-39).
23:2 The scribes and the Pharisees were two distinct groups, though there was some overlap between them: the scribes were the professional interpretative experts on the Torah itself, while the Pharisees were experts in theological matters that the Torah raised. Moses’ seat. Traditionally understood as referring symbolically to the authority of Moses. However, recent archaeological evidence has revealed a literal chair found in early synagogues. Whether literal or figurative, it refers to a place from which experts on the law taught.
23:3 so practice and observe whatever they tell you. Jesus recognized the Pharisees’ official function as interpreters of the Law of Moses, and insofar as they accurately interpreted Scripture, they were to be obeyed. However, “so” (Gk. oun) connects this verse with v. 2 and the mention of Moses, and therefore “whatever they tell you” should probably be limited to “whatever they tell you about the Law of Moses” and does not include the Pharisees’ later extensive additions to Mosaic laws which rabbinic teachers made. but not what they do. Jesus is about to show that much of the Pharisees’ practice and their extrabiblical tradition is wrong.
23:4 Heavy burdens describes the extrabiblical tradition of the rabbis that was a pillar of the Pharisaic branch of Judaism. It was intended as a means of making the OT relevant to new life situations, but its massive obligations had become burdensome and oppressive.
23:5 phylacteries. Small cube-shaped cases made of leather, containing Scripture passages written on parchment. They were worn on the left arm and forehead as a literal way to obey the admonition of Deut 11:18 (cf. Ex 13:9; Deut 6:8). fringes. Tassels with a blue cord that were attached to the four corners of a man’s garment (Num 15:37-41; Deut 22:12), reminding the people to obey God’s commandments and to be holy (Num 15:40).
23:6 place of honor. Seating at banquets was assigned to guests based on their rank or status. best seats in the synagogues. Excavations at early Galilean synagogues indicate that bench seats were built along the sides of the synagogue (see note on Luke 4:16; and The Synagogue and Jewish Worship). In any meeting place, some seats are regarded as better than others.
23:7 Rabbi (Hb. rabbi) literally meant “my lord,” but it was used generally for outstanding teachers of the law, most frequently heads of rabbinical schools.
23:8-10 not to be called rabbi,
..
. call no man your father.
.. Neither be called instructors. Jesus’ disciples should not try to gain authority over one another as teachers or masters, since Jesus is ultimately each disciple’s teacher and master (you have one teacher*.
.. one instructor), to whom the disciple is accountable. Jesus does not literally forbid use of the titles “teacher,” “doctor,” or “father” for all time in all circumstances, but he prohibits his disciples from using these terms in the way the Pharisees used them, in a spirit that wrongly exalted leaders and reinforced human pride.
23:13-36 Woes of Judgment against the Teachers of the Law and the Pharisees. Jesus now addresses the scribes and Pharisees directly, declaring a series of seven “woes” upon them that echoes the criticisms he has repeated throughout his ministry. These seven woes stand in contrast to the first seven “blessings” that introduce the Sermon on the Mount and describe Jesus’ true disciples (5:3-9). (For a similar list of woes, see Luke 11:37-54.)
23:13
First woe: the shut door. The woes are a mixture of condemnation, regret, and sorrow. The teachers of the law and the Pharisees are false leaders who have drawn the people away from the kingdom of heaven instead of toward it.
23:15
Second woe: entrapped converts. Jesus does not criticize proselytism per se, but the manner in which the Pharisees zealously sought converts, only to place them under the burdensome weight of the many requirements in their extrabiblical traditions. child of hell. Literally, “child of Gehenna,” a reference to the Valley of the Son of Hinnom, just south of Jerusalem, where refuse was burned. Jewish and NT writings used it as a metaphorical picture of eternal punishment (see note on 18:6-9).
23:16-18
Third woe: binding oaths. The Pharisees distinguished between oaths made by the temple and those made by the gold of the temple, and between oaths made by the altar and those made by the gift on it. As in much of their belief system, they focus on misguided superficial distinctions and overlook the higher principles of the law.
23:19-22 whoever swears. Those with faith in God who recognize their constant accountability in his presence need only give a simple “yes” or “no” as a binding oath (see 5:23, 34-37).