Neither of those systems should lag. I sounds more like a configuration error. Make sure that you have the computers power plans set to Performance. That forces the computer to use it's full capabilities. And make sure you are using Graphics drivers directly from the Nvidia support site and not from Windows Update. Also make sure that your system is only using the Nvidia graphics card. If you are trying to use the CPU's onboard Intel graphics it would probably introduce some lag. 

The 1070 is about 13% faster than the 1060. The 4060 is about 25% faster than the 1070, and the 4060Ti is about 14% faster than the 4060. If you want to upgrade the card, I would look at the price to determine whether the difference between the 4060 standard and 4060ti is worth the minimal performance difference. Personally I don't think the 4060Ti is worth the $180 US price increase over the standard 4060. $280 vs $450 just doesn't seem  reasonable to me.

As a side note, our church uses a 1660 Super with absolutely no issues. That card is within a couple of percent of the 1070 in performance.