Partly it depends upon your learning style, but also I'd note that part of the problem with the tactics site you mentioned isn't the computer. It's you.
Seriously, there's nothing stopping you from sitting and thinking about the position on the screen the same as with the position on a board. Nothing except you, that is. And in an age where more and more chess play will be online, it might be a good idea to use it to train yourself not to do that with positions on a screen.
Yusupov's rule is good; think about the position, figure the whole continuation out before you move, and if it takes you more than 10 minutes, either set it aside to come back to later or look up the solution. Just do that on the tactical site; if your personal style doesn't let you do that then yes, I'd say you shouldn't try using that kind of Internet site for anything other than entertainment. Doesn't mean it's bad; doesn't mean you're bad. Just means you two aren't suited to each other so you'd best agree to date other people (chess sites).
When it comes to learning chess, it's more important what you do than where (or how) you do it. build up a library of positions you understand, learn to recognize when that combination is lying under than thin coverlet waiting to be exposed. Understanding and pattern recognition are what's important, not the road you travelled to meet them.
I'm old school; I learn best when I'm holding the pieces in my hand, developing the muscle memory along with the chess memory. Shirov, OTOH, once said in an interview that he doesn't even own a chess set. I'd be a fool to say my way is better than his. I'm fairly certain I'd learn more from him in a hour without a board than he would from me if we spent all day at one.
For some people it's easier to recognize a pattern when they've trained themselves on it using the same kind of pieces as they would use in the game; they can even get thrown off by slight changes in design or size. Others could use silverware and condiment containers on a checkered tablecloth. Your best way forward is to learn what sort of learner you are, and leverage that. Maybe use some spare cycles here and there to expand your learning approaches, but in the main stick with what works for you. Less wasted energy that way.