OK one thing at a time. Online and Desktop program are separate and separately priced. You can't apply to Desktop payment to the Online version.
Yes you can transfer 2020 online return into the 2021 program. Don't know why you couldn't download the .tax2020 file. Were you in the right account? But the other way is to start a 2021 online return and transfer into that. Sounds like you did that so you can use the steps below.
If you already started a 2021 online return you can open it in the Desktop program without paying for Online. If you do finish and file from the Online version you will have to pay for the Online version.
You can download the .tax2021 data file without paying for Online and continue in the desktop version. First download the .tax2021 file from Online. How to Save the tax file
https://ttlc.intuit.com/community/saving/help/how-do-i-save-my-turbotax-online-return-as-a-tax-data-...
How to switch to the Desktop program
https://ttlc.intuit.com/community/choosing-a-product/help/how-do-i-switch-from-turbotax-online-to-th...
After you get the program reinstalled the first thing to do before you open your tax return is to update the program and install any state programs you had. Then open your file. So you first might need to start a fake return to be able to download the state program (go to FILE - NEW). Enter some basic information, then go to the state tab and download the state program. Then try to open your existing return, go to FILE - OPEN
