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What's wrong with private baptism?
So much. Let's begin with history.
Moses was instructed to construct a bronze levar to place outside the Tabernacle. This was visible to all that the priesthood must be made clean prior to entering.
Next at the temple there was a similar scenario which was used for more washing of priests (ablution).
The levitical baptism, the same baptism John gave and Jesus received in the New Testament, is a priestly baptism done, like at the Tabernacle and the Temple, done publicly for all to witness.
Further all baptisms (NT and OT) observed in Scripture are all public for all to bear witness. The only time prior to age of "everyone can make up their own denomination" where baptisms were "private" is when the roman church allowed priests to baptize children immediately after birth because of bad pregnancies/deliveries. Otherwise even back then the norm was in front of the church for all to bear witness.
In fact, even here in America, baptism records
because the sacrament is public acted as valid governmental records of citizens, birth certificate, evidence of name and inheritance etc. Because it was public, it verifiable by witnesses in the township.
....we can get into the theology of the rationale of public baptism. But if 4000 years of examples of its visibility isn't enough, it's gonna be a fruitless conversation.