Let's try to divide this into smaller problems and see if we can actually get an answer.
First, is it a problem with one particular spreadsheet, a few or with every one, created new or copied to it.
Since you say you have another working version of Excel, open a sheet on the non working computer, verify it doesn't work as described, exit Excel, copy exactly that sheet to your working computer, does it work there?
If it doesn't then the problem is more likely in the sheet. There are all kinds of things that can be done to a sheet, cells can be locked, whole sheets can be locked, and I think there are even more things that can be done to a sheet if someone is trying to create something that users can look at, but not ruin.
If it does work then the problem is more likely in the Excel on the computer that doesn't work.
Next, I am not a lawyer and don't know the precise wording of the current install policy for your exact version of Excel, but I've seen things to the effect that there are hard limits on the number of times you can install one copy of Microsoft Excel/Office, even on the same computer, before it begins to complain or limit functioning or refuse to even work. The reason I mention this is that if there are such hard limits then your "well I'll just install it a few more times and see if it is any different than the last few times" may be using up your quota. I've heard things like the quota is 3 times and that is it, period. Maybe you can apologize and beg and plead and see if Microsoft will override that.
You may or may not be running into such problems, you may or may not be making a much bigger problem for your self by randomly reinstalling this again and again. But I would be very hesitant to keep trying that, unless you have called Microsoft on the phone and hopefully had someone there who knows what they are doing tell you that it is just fine to keep reinstalling, as many times as you like, no problem.
Next, there used to be a "Am I Legit" web page at Microsoft for Windows and another one for Office that would check to see that your version was installed, registered, activated, blessed, etc, etc. The Office one appears to be down at the moment, but you might ask Microsoft Support to tell you what web page would let you check and verify that your current version of Excel on the non-working computer is all approved by Microsoft. If so then that should help minimize concerns about whether failing this is part or all of the problem. I'm pretty sure some Microsoft employee, even on the phone, has his whole job being just helping you check to make sure you are legit in your use of Microsoft Office products.
Now, for everyone else, what are the next two or three things he can do to divide and conquer and solve this? Something that will help narrow down specifically what his current problem really is? Something less than potentially just trading one problem for another by tossing Excel and starting over with a different brand?