All I can say at this moment is that both Windows defragmenter and Defraggler work fine here, I can launch and run either one as I choose.
Uninstalling my Defraggler does not affect Windows defrag in the slightest. Neither does reinstalling it.
You seem to have 2 seperate isssues here, the slowish startup, and then the Windows defragmenter not launching.
They are probably not related, but might be.
3 minutes for startup can seem a long time these days, but even only 5 years ago was often considered reasonable.
"Why is my computer slow to start" is one of the top questions asked online, and that's because there can be a multitude of reasons.
From a simple setting, through to too much bloat loading, system or hardware problems, up to even malware.
Have you had a look at just what is strating when you startup?
For example do you maybe have Edge pre-loading, even though you may never use it?
What about other background apps and bloatware?
I'd suggest running AdwCleaner from Malwarebytes to start with, it can quickly get rid of that junk/bloatware, adware, etc.
https://www.malwarebytes.com/adwcleaner
In windows Settings go to Apps>Startup and see what is turned on there that could be slowing things.
If you have things that you don't use turned on then turn them off.
You could then also look at Tools>Startup in CCleaner to see just what else is loading at startup, there are 4 tabs there.
You can also disable and enable them there if you are sure that you don't want them running at startup. (Don't use 'Delete' all it does is remove them from that list it doesn't delete them).
Next to the defragmenter.
As Hazelnut asked what type of drive is it? What size is it? How full is it? Right-clicking the drive in File Explorer and selecting Properties will tell you the Capacity and the used/free space on the General tab, the Hardware tab will give you the model number of the drive from which we should be able to tell what type it is.
While you are in the Properties go to the Tools tab and see if the defragmenter will launch from there. (And maybe run the Error checker too, it can take a while depending on the size of the disc, but would rule out another potential cause of problems).
You said that you were not happy with the Windows Defragmenter for various reasons, can you expand on those reasons just it case they might be relevant here?
Step by step, do you normally launch the Windows defragmenter? You mentioned from File Explorer above?
What do you see when it fails to load?
And finally for now, have you tried booting into safe mode and seeing if it runs there?