Macro that merges several excel files into one

Hello!
I tried to come up with a solution that can make my job easier but for some reason it's not very consistent. I will mention that I am quite new to the world of VBA and Excel but I'm trying to learn a few tricks, so bare with me please.

This is my problem: I have to add several separate excel files in one spreadsheet so that I can make a report on that data. Because sometimes there are quite a lot of files I tried to figure out a way to add them automatically. Every file basically looks like this:
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The files are saved daily and the usually have a filter active so I figured out a macro that does the following:
-first I have to get all the files I want to add into one folder
-then I open up a new spreadsheet, open VBA and paste my macro in and run it
-then the macro opens every excel file,clears the filters,selects every row of data except the first row,copies them in the new spreadsheet and closes the excel files without saving any changes.
-after it's done I insert the first row into my new spreadsheet and start working on the report.

So this saves up quite a lot of time but sometimes it doesn't work, it just adds the first row (the one with the First Name, Last Name, etc) from every excel file and no other row of data, and I don't really understand why.
Anyway, maybe someone can take a look at this macro and point out what's wrong, or even tell me if there is an easier way to do this.
This is the macro I came up with just by adding bits and pieces from the internet, so don't laugh pls.

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Thank you for your time and patience, and hopefully for your help.

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The logic appears to be OK. Do your sheets contain more that 65536 rows or gaps in column A?

No, not even close to 65636. There usually are close to a thousand. From what I can figure out there must be something wrong with the formatting of the excel files when the macro doesn't work properly. Or maybe with this part of the code, since it does exaclty the opposite, instead of deleting the first row, it deletes everything else.

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Later edit:
Ok, so I messed with it a bit and I think I found out why it isn't consistent all the time.
Some of my databases have no info in column A, there's just the name of the column in A1 and blank from A2 downwards. After I inserted some random data in column A the macro worked just fine.
Does anyone have an ideea how to fix this in the code? I'm not sure how to write an "if" command like "if rows are blank then copy them anyway" :D

Thanks anyway!