See it would need a long explanation to bring all the meta-points which I am not feeling inclined to now — it is something one could write a whole essay about but hardly my priority of things to write. but key points are it is clear from bAlakANDa
1st: tasya ca evam prabhAvasya dharmaj~nasya mahaatmanaH |
sutArtham tapya mAnasya na asiit vaMshakaraH sutaH ||
He is clearly having a burning for the son who will extend the line — an anxiety. He has 3 wives but has not procreated. This is clearly a sign of a problem. Old tradition allows in these circumstances for an appointee with virtue to play the role of the sperm donor
Next point: chintayAnasya tasya evam buddhiH AsIn mahAtmanaH |
sutArtham vAjimedhena kim artham na yajAmy aham ||
He now decides he should do vAjimedha for offspring. That is an unusual call for a horse-sacrifice. The horse sacrifice is normally done for a victorious king however we all know that the sexual rite at the heart of the ritual as well as the royal “fertilizing” yajuSh which is chanted to the horse implies that is always had a fertility angle which is what dasharatha is invoking rather than the martial one.
ashvamedha as fertility rite
Now, this has deep roots in the Indo-European world with parallel sexual connotations in the horse rituals in other tradiitons e.g. in Rome
That is also clear from the obscene yajuSh recited when the queen lies near the horse: ambe ambAlyambike na mA yabhati kashchana | sa sastyashvakaH | haye cha haye asau || Then in the ashvamedha the royal v2 women are given in hand symbolically to the v1 R^itvik-s and taken back after giving them the ritual dakShiNa-s. This again points to something of the deeper sexual ritual.
In the martial and standard ashvamedha it is only symbolic: but there is the horse phallus rite before that; all of this points to it once having a sexual significance that is only symbolic in the standard form. But note dasharatha clearly says he is performign the horse rite for offspring: an unusual but meaningful atavism.
mahAbhArata comparison
Now it is not explicit as in the mahAbhArata where vyAsa fathers the protagonists (stated clearly) but the parallel is important: 1. vyAsa fathers on ambikA, ambAlikA and sUtA(names parallel to the chant in the ashvamedha see above). 2. He asks for a 1 year vrata ike horse ritual but they don’t observe and the outcome is a disaster.
RShyashRnga
sumantra then specifically chooses R^ishyashriNga whose fertilizing nature like parjanya in the shruti (yo garbham oShadhInAm puruShINAM kR^iNoti…) is central to this tale: bring rain to the parched aNga. Now this same fertilizing R^ishyashR^i~Nga has to be brought to fertilize the barren “kshetra” of the ikShvAku monarch. That is the central motif there in his choice. This is then clarified by an interesting statement when romapAda sends his SIL:
shrutvA rAj~no.atha tat vAkyam manasA sa vichintya cha |
pradAsyate putravantam shAntA bhartAram AtmavAn ||
Note that the emphasis is not on the ritual confering the son but the husband of shAntA being the putravAn i.e. the son-conferrer. Now that to me is notable: the one officiant is specifically called son-conferrer rather than being the performer of the ritual
niyoga appeal
These are again almost mentioned as separate things when dasharatha humbly requests rishyashR^iNga: yajnArtham prasavArtham ca svargArtham ca nareshvaraH | for the sake of the ritual, for the sake of progeny and for the sake of heaven. Once the core horse ritual is done dasharatha says:
tato.abraviit R^ishyshR^i~Ngam rAjA dasharathaH tadA |
kulasya vardhanaM tvaM tu kartum arhasi suvrata ||
This to me is telling the rite is done he has given wealth to vasiShTha and RShyashRnga both of whom have done their ritual duties then he explicitly says you have to do my kulavardhanam
I see no reason why not take it in the literal sense (given all the other context presented before) and see it as the closest to direct admission that he performed kulavardhanam
In line with about note again the emphasis on RShyashRnga in addition to the ritual:
tad ahaM yaShTum ichChAmi shAstra-dR^iShTena karmaNA ||
R^iShi-putra-prabhAvena kAmAn prApsyAmi chApy aham |
He wants to perform rites ordained in scripture but also stress the “prabhAva” of the R^iShi-putra: this word can be interpretted in many ways but one way given the context is prabhAva as potency by which his desire of offspring is achieved
putrakAmeShTi
Now that part of the putrakAmeShTi comes with a major vaiShNava interpolation into what was originally an aindra epic. Originally he is only talking of hayamedha giving offspring but suddenly this atharvan rite is now brought out with the divine payasam a euphemism for the fertilizing semen similar to the ghee of pururavas in the RV. This to me is the attempt to euphemize and introduce divinity into the narrative.
Infertility of dasharatha
Having laid this base stuff from R bAla we come to your points: Was dasharatha infertile: we do not know if he was infertile, having erectile issues or a more subtle immunological issue. These are part of the human condition and the text is clear that he was anxious about not having offspring and despite having 3 wives not having them is something to keep in mind.
Now if the hayamedha was sole for removing his sins there is no such explicit mention that he desired his sins to be removed: he says he wants to do it for offspring.
Biology & historicization
Now if you want to historicize it you have to be tethered to biological realities; hence, when coupled with dasharatha’s condition we have to come to the reading as above of a direct agency of R in generating the offspring
Conclusion
What I will not go into is all the meta-information. So when we opt for a historicized reading then this role of rishyashR^i~Nga falls out as I’ve stated.