Excel Reads "" as Greater than a Number Value and it is MESSING UP My Formulas

I tried to produce your results with a simple =IF(A4="","",1) in A3 and got the same outcome in the IF/OR statement.
So instead I put this in for the A3 =(A4="",,1) and it produced a 0 but the output for the IF/OR formula worked ok.
Maybe seeing your formulas in A3 (and B3 etc if they are different) could help someone develop a formula so your output works in your IF/OR formula.

As you note, a text string (even the null or empty string "") is always "greater than" a number.

What is the null string doing here? Are you just using it out of habit? Could you return 0 (or other clearly meaningless but small number) instead? How about an NA() error (that would readily propagate through the dependent formulas)?

I could see adding an additional IF() condition that tests if there are any numbers in A3:D3 =IF(COUNT(A3:D3)=4,current formula,0). The COUNT() function will ignore the text, so it will wait until all 4 cells have a number before calculating the final result.

A lot depends on what you are trying to do with the null string and how that fits into the rest of the logic of the spreadsheet.

is it out of the question to change this to zero .... =IF(Portfolio!E3>0, DAYS(TODAY(), Portfolio!E3),0)
if you don't like zeros showing up you can always use conditional formatting to make them the same font color as the background fill color.

Well the obvious other thing to do is to slap additional terms in the Boolean logic that check
ISNUMBER(E3)
etc.