Matching a character in a range of strings at specified positions?

I'm wanting to check an entire range of strings in one go, to see if they contain the letter Y at specific positions decided by the user.

I have data in the format:

Y____Y_Y_Y
_Y__Y_____
Y_Y_Y_Y___
...

These are all Strings of the exact same length, but the Y's and _'s can be in any position.

I wish to check these Strings and see if there are Y's at specific positions. I have a tickbox on whether to check that position or not.

I then want to check if the letter Y would appear at position 1, 3 and 6 in this example.

I am able to check one line at a time to see if they contain a Y at any of the specified positions using the formula:
=IFERROR(FIND("Y",ArrayFormula(JOIN("",IFERROR(MID(D2,$F$2:$F,1),))))>0,FALSE)

D2 contains the single string to check.
$F$2:$F contains the extract positions to check.
The MID takes all the letters at the specified positions, which is then joined into 1 string.
This string is then checked to see if it contains any Y's, returning TRUE if so.

My problem is that rather than dragging down the formula onto the cells below, so it checks each string individually, I wish to do this in one line. Simply changing the MID formula to be a range for D2:D25 for example does not work, as I believe MID cannot check both multiple strings and multiple positions at once.

Is there a way to extract letters at specified positions and check whether they contain a letter "Y" from an entire range?

Provided test spreadsheet for reference

The reason your formula doesn't work it's because of the JOIN function not the MID function. If a function takes arrays by default, it won't work as you would expect when used with ArrayFormula.

Try this instead:

=ARRAYFORMULA(IF(D2:D="",,0<MMULT(N("Y"=MID(D2:D,TOROW(F2:F,1),1)),N(TOCOL(F2:F,1))^0)))

You may try:

=map(D2:D,lambda(Σ,if(Σ="",,countif(index(mid(Σ,tocol(F2:F,1),1)),"Y")>0)))