Hi and welcome
My wife'car is white and I'm a bit of a detailer myself

As you say white does show up tar spots and Tardis (which I also use btw) is quite harsh.
I've found that some tar spots will wash/polish off if you keep your car well waxed/sealed.
However some other tar spots are a PITA to get rid of, and Tardis is needed. It really seems quite random ... some months hardly any spots appear, and then suddenly it can get coated in them. Hopefully you've been unlucky and just had one of the latter

I have resigned myself now to putting up with some of the tar spots and only giving it a full Tardis treatment every 6 months, followed immediately with a good wash, rinse, polish and wax.
For the odd few stubborn spots try a bit of Autoglym Super Resin Polish or a dab of Tardis on a cloth