This may not be an answer but too long for a comment!
Though I did not read your mentioned paper https://www.ams.org/journals/proc/1972-036-02/S0002-9939-1972-0334212-5/S0002-9939-1972-0334212-5.pdf in details but what I understood from the Peter May's answer here Homotopy of functors that if you define a homotopy between 2 covariant functors $F,G :C \rightarrow D$ as a natural transformation between them then this notion of "homotopy" will not induce an expected equivalence relation on the set of functors from $C$ to $D$. So to solve this problem Ming-jung Lee defined a notion of "homotopy" between $F$ and $G$ as a sequence of covariant functors $\phi_1,....\phi_n:C \rightarrow D$ such that $\phi_1=F$ and $\phi_n=G$ and such that for each $i$ there exists a natural transformation between $\phi_i$ and $\phi_{i+1}$(where the direction of each natural transformation is unspecified). Also as mentioned in corollary 8 in https://www.ams.org/journals/proc/1972-036-02/S0002-9939-1972-0334212-5/S0002-9939-1972-0334212-5.pdf that if $F$ and $G$ are "homotopic" then the induced continuous maps $BF$ and $BG$ between geometric realisations $BC$ and $BD$ of the morphism complexes $MC$ and $MD$ respectively are also homotopic. So the notion of "homotopy" as mentioned by Ming-Jung Lee seems reasonable..