In fact, I have been troubled by a similar problem for a long time. Of course, I can't determine whether this problem is caused by too many uninstalls, but I can tell you clearly: usually this is caused by incomplete uninstallation or incomplete installation. If you know your macOS well enough, you should know that FortiClient is not just a desktop application, it also generates many services in your system, and these services may have irreparable errors. Of course, in addition to completely restoring the system, we might as well try the following. If you are lucky, the problem may be solved.
S1:Uninstall FortiClient completely, I would recommend you to use "FortiClientUninstaller.app", of course, you can also use a third party software, such as CleanMyMac;
S2:Reboot;
S3:Open Terminal.app and type "launchctl print-disabled system", you must record all the listed especially which contains fortinet;
S4:Next, the most critical step, for each record above, you should continue to type "sudo launchctl enable system/com.fortinet.xxx", then enter your password and hit Enter; make sure you remember to run this command for each one of the above;
S5:I would suggest you to reboot quickly;
S6:Re-find your dmg or zip installer and install FortiClient completely from scratch;
S7:All Right, if you're lucky enough, maybe when you open it again, you won't see the white screen anymore and everything is back to normal.
Sincerely hope I can help you with this disgusting problem, buy me coffee if you want!