TLDR: Putting stones on the board is not always possible. Reading is super important. Practice it with tsumego and apply it in games.
You're definitely right that it has a strong impact on the outcome. Reading is the most fundamental skill in Go, and improving in that area is absolutely required to become a strong player.
In serious games, it is usually not possible to put stones on the board "just to see what happens". You'll have to imagine the results in your mind before you put down a stone. This is a fact that can't be changed.
However, as a beginner, you should try some stuff by yourself or in non-serious games (with knowledge of the other player, obviously) first. Then, it is absolutely fine to put the stones on the board to check what happens. You'll then quickly become aware of typical shapes (for instance a ladder), and be able to spot it immediately (if not, just keep trying!), without the need to put actual stones down. This is the first step on the road to strong reading.
In teaching games, you'll usually also lay down important sequences (during or after the game), in particular when they are not easy to see. Try to remember them so you'll notice them next time!
Further, you will need to improve your reading abilities. Typically, players practice solving tsumego for this purpose, as they require reading skills of a particular level (yours in this case). First make sure you really solved the problem - do not miss a strong, difficult-to-see reply by your virtual opponent. Then, to check your solutions, it is fine to put down stones to make sure you did not miss anything. It is also beneficial to have reliable solutions at hand with which you can compare your own, for instance by using a tsumego book.
Over time, tsumego as well as simply practicing reading in real games will improve your abilities and you will be able to read faster, deeper and more precisely. In particular, as you memorize the involved shapes, you'll immediately know which moves are impossible without considering them. This will drastically speed up the selection process and cut down the variation tree to very few branches, each of which you have to explore as best as your visualization abilities allow you to.
As a beginner, it is actually often sufficient to always read only 2 or 3 moves ahead to avoid stupid mistakes. Once this does not help you anymore, try reading a few moves more, and you'll certainly do more than most lazy dan players :)
This last tongue-in-cheek sentence is actually true - when they are not serious, advanced players often rely on shape exclusively without reading much, as they can already tell the results from the initial shapes (often, but not always correctly). This ability is beneficial in very fast games, but in serious games you still need to actually read (deeply) to find the best moves. Use memorized shapes only to reduce the tree width.
By the way, aspiring pro players typically spend a few hours every day on tsumego. It is, after all, said that pros solved every tsumego in existence at least twice. If you enjoy it, feel free to do alike! If not (which applies to many people), well, just do it if you feel like it - Go should be fun! Tsumego are often more fun when hand-picked and given by a friend, possibly to many people at once, with a joint resolution session which typically includes a lot of "Oh!"s and "Aaah!"s.
Regarding "visualization" of moves. I did not go into that point yet. This is something I kind of wonder about myself, as I found it difficult to see clearly the moves very deeply, in particular when "new" stones touch stones from a while ago or are even played on top of them (this is called "ishi no shita" or "under the stones"). As far as I can tell, you cannot work on this specifically - it's simply a result of studying and will hopefully improve as one gets better at tsumego.
As a final, practical advice: Copy or print out tsumego from a good book on cards or cut little paper sheets. I liked this a lot, but definitely start with the easiest book, as the problems are often harder than they appear at first! Then put them in various places in your house as well as in your pocket. Every time you are forced to wait for a minute or more, pick one of them and try to solve it. Repeat until the wait is over. I often did that in waiting rooms (even though I have to admit that it sometimes made me rather sleepy).
Quoting tasuki on the above works:
Cho Chikun’s Encyclopedia of Life and Death is a wonderful progression which will take you from an absolute beginner to mid dan level.
Next, we present you with the three most famous classical tsumego collections. Gokyo Shumyo deals mostly with patterns occuring often in real games. XuanXuan Qijing focuses on feeling and vital points, and the insanely difficult Igo Hatsuyo-ron is here just to make you cry.
They come without solution due to copyright, but you can mostly check them with friends and/or stronger players. By the way, I can absolutely confirm his statements, in particular about Igohatsuyoron :'(