Hi, I would like to start off by saying that I am no way an expert in aquarium plants or ferts, but this is I experienced going from no dosing to dosing.
When I first started my tank I used florite but no added ferts. All of my plants grew fine no problem with the exception of a few of the more advanced types of plants I purchased. The plants that grew best were things like swords, crypts, and java fern.
A few months into it and I built a diy co2 system (which I eventually changed to a real co2 system). After adding this I noticed a lot more growth in my plants.
I have recently started dosing with Seachem flourish, excel, and iron. I feel like I have seen a little more growth, though its a little to early to tell, but I have definitely noticed my plants just overall look much better and healthier. The leaves seem to be much fuller with great color. I plan on starting to use phosphorous, potassium and nitrogen soon. I only dose 2x per week with the iron and excel instead of the recommended every other day. Mainly just to save money though, I can't afford to run through the stuff to fast.
So to answer your no you do not necessarily need ferts with good lighting and easy to grow plants, but it does make a BIG difference if you do use them. This will be especially true with a shrimp tank since they do not produce the waste for the plants to use like fish would.
On a side note you can buy a seachem pack with small bottles of flourish, excel, and iron off amazon for just 10 bucks which would last a while for a nano. You can also get the fluval co2 nano kit for around $25 on amazon. I ordered a cheap $5 diffuser bc I didn't like the one it came with. Also you can order 16g threaded co2 cartridges off ebay for around $1 each instead of paying $15 for 3 of the 20g fluval brand ones. Sorry for such a long post, but hope it helps.