I think the simple answer might be that if the internet connection adds value for the player... some players will at least give it the benefit of the doubt. I think you'd have more luck with something silly or fun, rather than functional.
Some will still avoid it, just to be ornery. Larger publishers have totally destroyed some player's faith in what their internet connection might be used for... and sadly, you have to live with the consequences. The rest just don't understand how their data is used.
As long as you give players the choice of whether to participate or not, you might get away with it. Especially if there's an obvious advantage to taking part.
If however, you move something to your web interface that could just as easily have been displayed in game... not so much.
Daily, weekly and/or monthly leader boards could be seen as offering an added dimension to your game... depending on your game's mechanics. If you end up using any sort of geo-location type stuff... let the client do it and just send a country code to your server... Nothing freaks some people out like seeing their own IP address sent as data.
Forget anything involving time or speed... Someone will just hack things and you'll ended up with to top-50 leader board being 0.086 seconds or some other such rubbish.
Also keep in mind you might end up getting hacked. So don't collect any information you wouldn't be comfortable explaining to your userbase that someone grabbed everyone's data.
Finally, whatever you do... keep in mind that your game needs to continue to work, even if you pull the plug on your web interface. There's nothing like paying $40/mth** for webhosting and earning $35/mth from Patreon to really make you question why you added the online component. Especially 2 years down the line...
** I pulled that $40/mth for webhosting out of the air. I know it's not a real number, I was just using it to illustrate a point. No need to tell me it's unrealistic or list your favorite webhosting company that do hosting really cheap.