I've found something weird.
Lectus is a bed.
In French, a bed is called a "lit".
Lectus also mean something that is selected, or read.
In French, "lit" as a verb means "read(s)".
So, it means that the lectus meaning a bed gave in French a "lit",
and the verb lectare also gave "lit" in its conjugation.
For lectus, it would be "lu" in French, but it's still a weird "coincidence"?
Lectus 'bed' has a short e; lēctus 'chosen', 'read' has a long e. Lit is the expected reflex in French of Latin lectus and Latin legit, but lu is not the expected reflex of lēctus; French lu should come from a Vulgar Latin legutus, an analogical regular participle of legere; Spanish also has leído, a regular participle, instead of the expected lecho < lēctus.
I don't think the verb would be 'lectare'. Verbs in Latin are usually named according to their present indicative first person, which in this case would be 'lego'. Among other meanings, it can mean 'read' or 'choose'. The corresponding infinitive, which is what you were looking for re 'lectare', is 'legere'. Compare to French 'lire'.
For comparison, the Latin infinitive is very close to the Italian current form 'leggere' (read) and 'eleggere' (elect / choose), or Portuguese 'ler < leer < legere (lt)' (read) and 'eleger < eligere (lt) < ex + legere' (elect / choose)
I'd say French is a bit more removed from Latin, so much so that you will pick coincidences from hitting on the same derived form coming from different sources over longer derivation routes.
As for lectus in the sense of bed, it became 'letto' in Italian but the 'c' was vocalized in Portuguese, in a rather common derivation, to become 'leito'.