Can 1 be written as a finite sum of distinct unit fractions from any arithmetic progression?

Let $a,d$ be positive integers. Is there a reasonably short elementary proof that one can find distinct nonnegative integers $n_1,\dots,n_k$ such that

$$ \frac1{a+n_1d}+\frac1{a+n_2d}+\cdots+\frac1{a+n_kd}=1? $$

Equivalently, can one choose finitely many distinct terms of every infinite arithmetic progression of positive integers so that the sum of their reciprocals is $1$?

I know that this is a special case of the theorem that every infinite arithmetic progression is a reciprocal basis, proved by P. J. van Albada and J. H. van Lint in Reciprocal Bases for the Integers, Amer. Math. Monthly 70 (1963), 170–174. I am looking specifically for a self-contained proof using only elementary number theory, preferably one that could be presented to strong olympiad students.

There is an easy proof under the additional assumption $d\mid a$. Write $a=cd$. Since the harmonic series diverges, choose $m\ge c$ maximal such that

$$ S=\frac1c+\frac1{c+1}+\cdots+\frac1m\le d. $$

Then $0\le d-S<1/(m+1)$. By the usual greedy Egyptian-fraction algorithm, $d-S$ is a finite sum of distinct unit fractions whose denominators are greater than $m$. Thus

$$ d=\sum_i\frac1{q_i} $$

for distinct integers $q_i\ge c$, and after division by $d$,

$$ 1=\sum_i\frac1{dq_i}. $$

All the denominators $dq_i$ belong to the progression $$a,a+d,a+2d,\dots$$

The difficulty is to remove the condition $d\mid a$.

A tempting identity is

$$ \frac1{cd} =\frac1{1+cd}+\frac1{cd(1+cd)}, $$

because the first new denominator is $1\pmod d$, while the second term has the same form as the original one. Iteration, however, gives only an infinite expansion: after every finite number of steps a positive tail remains. Other local splitting identities that I tried produce repeated denominators.

Is there a short constructive proof of the general case? In particular, is there an elementary way to guarantee both finiteness and distinctness without using Dirichlet's theorem or analytic number theory?