Efficient way to collate/aggregate specific data in Google Sheets - google-sheets

Question

I'm looking for an efficient way to gather and aggregate some date in Google Sheets. I've been looking at the query function, pivot tables, and Index + Match formulas, but so far I've not found a way that brings me to the result I'm looking for. I have a set of data which looks more or less as follows.
The fields with an X represent irrelevant data which I don't want to show up in my end result. They only serve to illustrate that there are columns of data that I don't want in between the columns of data that I do want. The data in those columns is of varying types and of varying values per type, they are not actually fields with an "X" in it. Only the fields with numbers are of interest along with the related names at the top and left of those. The intent is to create a list that looks more or less like this.
I've highlighted those yellow fields because that data has been aggregated. For example, in the original file field D3 shows a relation between Laura and Pete with the number 1, and field L3 also shows a relation between Laura and Pete, so the number in that field is to be added to the number in the other field resulting in an aggregated total of 2 for that particular combination.
I would really appreciate any suggestions that can help me get to an elegant and efficient solution for this. The only solutions I can come up with would involve multiple "in-between" sheets and there just has to be a better way.
UPDATE:
Solved by applying the solution in player0's answer. I just had to switch around the order of Col1 and Col2 in the formula to get the table sorted the way I needed it. Formula looks like below now. Many thanks to both player0 and Erik Tyler for their efforts.
=INDEX(QUERY(SPLIT(FLATTEN(A2:A&"×"&D1:N1&"×"&D2:N), "×"),
"select Col2,Col1,sum(Col3)
where Col2 is not null
and Col3 is not null
group by Col2,Col1
label sum(Col3)''", ))

Answer 1

try:
=INDEX(QUERY(SPLIT(FLATTEN(A2:A&"×"&D1:N1&"×"&D2:N), "×"),
"where Col3 is not null and Col2 is not null", ))
update:
=INDEX(QUERY(SPLIT(FLATTEN(A2:A&"×"&D1:N1&"×"&D2:N), "×"),
"select Col1,Col2,sum(Col3)
where Col3 is not null
and Col2 is not null
group by Col1,Col2
label sum(Col3)''", ))

Answer 2

Given your current data set (which only appears to extend to Col N), place the following somewhere to the right of Col N:
=ArrayFormula(SPLIT(TRANSPOSE(QUERY(TRANSPOSE(QUERY(SPLIT(QUERY(FLATTEN(FILTER(IF(NOT(ISNUMBER(D2:N)),,D1:N1&"~ "&A2:A&"|"&D2:N),A2:A<>"")),"Select * WHERE Col1 Is Not Null"),"|"),"Select Col1, SUM(Col2) GROUP BY Col1 LABEL SUM(Col2) ''")&"~ "),,2)),"~ ",0,1))
It would be better if this were placed in a different sheet from the original data. Supposing that your original data sheet is named Sheet1, place the following version of the above formula into a new sheet:
=ArrayFormula(SPLIT(TRANSPOSE(QUERY(TRANSPOSE(QUERY(SPLIT(QUERY(FLATTEN(FILTER(IF(NOT(ISNUMBER(INDIRECT("Sheet1!D2:"&ROWS(Sheet1!A:A)))),,Sheet1!D1:1&"~ "&Sheet1!A2:A&"|"&INDIRECT("Sheet1!D2:"&ROWS(Sheet1!A2:A))),Sheet1!A2:A<>"")),"Select * WHERE Col1 Is Not Null"),"|"),"Select Col1, SUM(Col2) GROUP BY Col1 LABEL SUM(Col2) ''")&"~ "),,2)),"~ ",0,1))
This separate-sheet approach and formula allows for the original data to extend indefinitely past Col N.