Evaluation of a synthesis from aspirin to paracetamol from this Nile Red video
You are absolutely correct that $6~\%$ yield is not acceptable for a synthesis — especially if it is a reaction as simple as the nitration of phenol. A quick search in SciFinder gives numerous reaction conditions that allow for up to $98~\%$ yield of the para-substituted product, depending on which catalysts and sets of conditions you want to use. One of the simplest sets of reagents is probably Ducry’s and Roberge’s Angewandte paper which, however, uses a microreactor. While their best selectivity is $1.2:1$, their yields are way above $12~\%$ (which would be the corresponding yield for $6~\%$ of the para-product in a $1:1$ mixture).[1]

Having an acetate group rather than a hydroxy group greatly modifies the properties of that particular oxygen atom. I synthesised a phenol and the corresponding acetate and the chemical shift of the ortho carbon changes by $7.5~\mathrm{ppm}$ upon acetylation while the meta carbon remains essentially unshifted. Therefore, I wish to challenge your assertion that acetate be ortho and para-directing — please back this with appropriate references. However, I will acknowledge that an acetate group is less meta-directing that a carboxylic acid; therefore the preferred reaction site should still be ortho/para to the acetate.

Whatever the selectivity, I do not agree that nitration of acetylsalicylic acid or salicylic acid would be more efficient than nitration of phenol. Remember that the reaction you are trying to perform is an electrophilic aromatic substitution $\mathrm{S_EAr}$. For an aromatic species to be attacked electrophilicly, it should have a high electron density. The carboxyl and acetate groups are both electron-withdrawing and therefore hinder the reaction. It should be harder and require harsher conditions to nitrate a benzoic acid than a phenol. The choice of steps is therefore better in the originally proposed way, in my opinion.

I will agree that decarboxylation of a meta-nitrobenzoic acid should be easier than of salicylic acid. Since the yield of carboxylation is already rather acceptable already, I do not think that switching the order would provide any significant advantage here.

Finally, it should also be rather easy to increase the yield of the final-step acetylation. $18~\%$ needs a lot of work.



Reference:

[1]: L. Ducry, D. M. Roberge, Angew. Chem. Int. Ed. 2005, 44, 7972. DOI: 10.1002/anie.200502387.