I think of fertilization in an aquarium as a way to control deficiencies.
If there is no apparent deficiencies, I don't add any. A some point the water will be depleted from some elements, deficiencies will appear, while some other compounds are building up.
With water changes and good dosing. I'm able to grow easy plants that would never made it before.
I use a complete fertilizer with Co2 included, formulated for "low-tech aquariums" over a year I probably use less than 1/20 of the prescribed dose.
But still, without it, I would never have an anubia sprouting a leaf in a month. Since I use it as soon as there is a slowdown or new leaf looks like having problems. Now they all crank a leaf per week.
So, on the algae control, the effect is more indirect, If the plant grows they are the one that are making a difference. If the plants are not growing and there is enough competition it can turn into an epic algae bloom.
Fertilizer will encourage plants. Decaying plant matter, food and fish poop will encourage algae. Both are good for plants, but most of the time algae can thrive with an incomplete diet, many plants can't