I just read up on the background on this, and wow, what a weird quirk of game versioning.
To directly answer your question: Removing the game from your account and repurchasing the Standard edition should pick up the installed game files correctly. The Standard and Starter editions of the game are listed under the same Store Page entry, they have the same App ID, so you should be able to do this without having to uninstall/reinstall the game files, although you should take a backup of the game folder under steamapps first, just in case.
See this answer for instructions on how to remove the game from your Steam account.
As to whether or not this is the correct way to go to 'upgrade' your account is up to you, but I thought it worth mentioning that there are other options. This Reddit Thread seems to concur with your solution, however another potential solution is to Purchase the Standard edition through U-Play. Your U-Play account will show 2 separate entries for the game, but launching your original Steam version will launch the game with the Standard edition stuff enabled.
Finally, purchasing the operators directly with R6 credits appears to be the officially supported solution, although the community seems split on whether this is a good cost/benefit tradeoff when compared to the above options:
Can I easily upgrade to the Standard Edition?
The easiest way is to gain access to all the Original Operators as quickly as the Standard Edition is to purchase the R6 Credits bundle that best fits your needs, and unlock the missing Operators.
For instance, getting the 4920 R6 Credits bundle allows you to instantly unlock both the Attackers Bundle (all 10 original ATKs) and the Defender bundle (all 10 original DEFs) plus the additional content of your choice.
