I believe I have formulas that will work for you.
In Column J for the max for each row I used:
=XLOOKUP(MAX(B3:I3),B3:I3,$B$2:$I$2)
In the max rows, Row 13, 27, etc. I used:
=XLOOKUP(MAX(B3:B12),B3:B12,$A$3:$A$12)
Both of these formulas will result in #N/A if there is no data in the Row/Column. That is good. The MODE formulas will replace those with "" so that they will not be counted.
For the column mode shown in L3 in the screenshot use this formula:
=LET(range,TRANSPOSE(HSTACK(B13:I13,B27:I27,B41:I41,B55:I55,B69:I69,B83:I83)),INDEX(range,MODE(IFNA(MATCH(range,range,0),""))))
For the row mode shown in N3 in the screenshot use this formula:
=LET(range,VSTACK(J3:J12,J17:J26,J31:J40,J45:J54,J59:J68,J73:J82),INDEX(range,MODE(IFNA(MATCH(range,range,0),""))))