I wish to look at the theological reasons for why the WH/Nestle school could weaken the relationship between Father and Son by not saying God was manifested in flesh at the incarnation.
[Verbatim from OP]
Firstly, the statement in the Textus Receptus is wholly compatible with other scriptures, to name but two :
John's gospel expresses (in the literal Greek and with the actual Greek word order) "... God was the Word ... and the Word became flesh".
John's first epistle expresses (in the literal Greek) "... the life the eternal was with the Father ... and was manifested".
If God was the Word and Word became flesh ; then, indeed, God was manifested in flesh.
And if the eternal life which was with the Father (and since 'Father' is mentioned, then the life must be that of Son) was manifested ; then, indeed, God (the Son) was manifested in flesh which was seen and handled.
The extensive argument to retain Theos rather than hos and the extensive research into the horizontal line abbreviation involved is amply covered by Dean John Burgon in his many presentations in serial form in the circular of his day and published also, in lesser form, in his book Revision Revised to which @Andrew Shanks refers in his concise answer to the question quoted by by the OP.
Then, what is the force of 'God' rather than 'who' in 1 Timothy 3:16 and is there a weakening of doctrine if 'who' is adopted ?
Firstly, the force of the context is weakened.
The context begins with 'These things write I unto thee' : it is emphatic, it is authoritative, the epistle preempts the coming of the apostle in person, but if not then the epistle emphasises Paul's authority in absence.
Then, what is emphasised is the presence of God : in the house of God and in the church of the living God.
And subsequently Paul declares - the mystery of godliness.
To say 'who was manifested' is a hiatus. It is disjointed. It is uncontextual. It is unconstructive.
Nor has it any sensible antecedent.
Nor is it grammatical.
It is, in and of itself, a weak statement. And thus, as an expression of doctrine, it is weak.
But more importantly, the concept being expressed (if we adopt hos and say 'who') is that the 'mystery of godliness' was 'manifest in the flesh'. For the only other antecedent in the context is 'God'. Then if 'God' is the 'who' why not say 'God' ?
What purpose would there be in stating the pronoun, if God is the antecedent ?
And if the pronoun refers to 'God' then what harm can there be in re-stating the antecedent, in any case ?
But if the antecedent of 'who' is 'the mystery of godliness' then we are left with a meaningless statement.
In a vague kind of way, some are suggesting that 'Jesus' (my quotes emphasise that this is an improper way of referring to Jesus of Nazareth, who - now risen and ascended - is titled 'Jesus Christ' or 'Lord Jesus Christ') is 'the mystery of godliness' which could mean no more than that he is a human who was an example of godliness.
And thus is lost any relationship of Son to Father. Lost is any relationship of the Christ to God.
It could be inferred, from this passage on its own, that Jesus was an example of how to be godly and he was no more than that.
But that is not what the Textus Receptus says.
The TR says 'God was manifest in flesh'.
The reverberations of which are immense. And which are re-iterated in John's gospel and in John's first epistle.