Whoever told you Esperanto lacks verbal aspect was lying to you. Yes, aspect isn't mentioned in the 16 Rules. However, this clearly doesn't mean Esperanto completely lacks aspect -- the 16 Rules are not intended to be a linguistically rigorous analysis of Esperanto, but merely a set of easy rules to teach laymen. Zamenhof definitely designed Esperanto containing some aspectual markers, especially given the amount of influence he took from Slavic languages; what is unclear is exactly how much aspectual information he intended to be encoded in certain markers.
Based on your description of Esperanto's complex tenses and your claim that "these aspects do not even occur in a pure form in English", you seem to be under the impression that to truly be present in a language, an aspect must be marked by a single morpheme rather than through syntactic constructions like the verb + participle constructions in English and Esperanto "complex tenses." However, that isn't how that works. The English present/past continuous constructions do mark progressive/continuous aspect. The English present/past perfect constructions do mark the perfect, though "perfect" is a bit more of a complex mix of tense and aspect than it is pure aspect. The fact that certain syntactic constructions are used here rather than a single affix does not make the aspectual marking here any less "pure".
There actually has historically been some debate over how Esperanto participles mark aspect when used in complex tenses. Zamenhof himself saw Esperanto participles as marking a mixture of tense and aspect, and this resulted in a divide among Esperantists regarding whether to translate, say "The house was built long ago" with estis konstruata (because the participle happens at the same time as the main verb) or with estis konstruita (as the building is accomplished and focused on describing its result). The latter group largely won out with an un-official notice from the Academy in 1965, which gave the following positions:
- The Esperanto tense system consists of but three tenses: the present, the past, and the future.
- All other time relationships (such as the pluperfect, future perfect, etc. of many other languages) are expressed by adverbs, conjunctions, or simply by context.
- The six participles, used either with nouns or with the verb esti, show aspect, not tense. They show, in other words, in what phase of the action the subject finds himself with respect to the object: whether beginning and unfinished (-ant, -at), finished and fulfilled (-int, -it), or not yet begun but intended or awaited (-ont, -ot).
- Accordingly the forms with -ata focus upon the duration or repetition of the act, and the forms with -ita focus upon the result of the act. The forms with -ita can also show priority in time in the event that there is separate evidence of that in the sentence, such as a word like jam = “already.”
(Quoted from Being Colloquial in Esperanto's Appendix on Participles, which goes into more detail about the conflict here.)
While this treatment of Esperanto participles are markers exclusively of aspect is technically not an official order from the Academy, it has generally been treated as such my most Esperantists, and this is largely how Esperantists use participles -- the -anta/-ata participles marking the continuous/progressive, the -inta/-ita participles marking the resultative, and the -onta/-ota participles marking the prospective. This is consistent with how they're used outside of the complex tenses as well. Based on this, one would have to willfully ignore a lot of Esperanto usage and convention to claim its participles don't mark aspect.
In addition to the obvious aspect marking on the participles, it should also be noted a few other affixes mark what could be considered aspect. You've mentioned ek- as a marker of inchoative aspect yourself (it could also be said to mark the momentane). -iĝ- could also be said to have some inchoative flavor to it (though it's a bit more obscured among its other functions, consider that eksidi and sidiĝi are near-synonymous), and -ad- similarly can mark something like continuative or iterative aspect.