Adobe Export PDF is an online service it can't be downloaded.
Make sure that you are "Signing In" using the correct email address under which you have made the purchase and you have installed the correct application Adobe Acrobat Reader DC not the Adobe Acrobat DC.
For further troubleshooting you may refer- https://forums.adobe.com/docs/DOC-8841
Let us know if you still experience the same problem.
How do I get the Icon on my new computer?
As per the description above, you want to install Adobe PDFPack as an application and want to get the icon on the machine, Is that correct?
Adobe PDFPack is an online service which cannot be downloaded as an application. But you can access the services through a browser and can be accessed through desktop and mobile app Adobe Reader DC. Use the services through the URL https://cloud.acrobat.com/convertpdf with your Adobe ID and password on a browser like Chrome or Firefox or Safari.
You can also use the free desktop version of Adobe Reader DC to use the services of Adobe PDFPack, just "sign-in" with your current Adobe ID and password. Download Adobe Reader from https://get.adobe.com/reader/enterprise/
Once the Adobe Reader is installed, you will get the icon of Adobe Reader on the computer desktop.
For more troubleshooting, you may refer to Troubleshooting Create PDF, Export PDF, Adobe Send, Fill & Sign … services(PDFPack or Document Cloud...
Feel free to update this discussion for any further assistance.