Close, but no see-gar, Kathleen. What you say is indeed true for the Eastern Orthodox and since 1920 the Anglican Communion and the Utrech Union of Old Catholic Churches, following the theory for transmission of valid orders set by St. Cyprian (who was arguably somewhat tainted by Donatism.)
It is different for the Catholic Church, which still officially follows the more mechanistic criteria of St. Cyprian’s opponent, St. Augustine. As far as the Catholic Church’s Canon Law Code is concerned, as long as the specific Augustinian criteria are met, an ordination is automatically valid, though it may be either “irregular” or “valid but illicit” (illegal.) In those cases, the Eucharist and other Sacraments may be irregular of illicit / illegal, but are nonetheless technically quite valid.
In some cases, (unless barred by a bishop within his own diocese,) as specified by the terms of Canon 844 of the Catholic Church’s Canon Law Code, there instances when Catholics may legally / llicitly receive some sacraments - especially the Eucharist and the Anointing (for last rites) from priests and bishops whose ordinations or valid but either irregular or illicit. There are specific limits for that however, which must be observed; and any bishop may not permit Canon 844 to apply within his diocese. This gets into an area of Canon Law which may be sometimes controversial.
Many of the Anglicans being re-ordained upon entry into an ordinariate are being "conditionally- “sub conditione” re-ordained rather than re-ordained “de novo,” because in addition to their Anglican ordinations, they have been ordained validly but illicitly / irregularly by Dutch Old Catholicand/or Polish National Catholic Church (PNCC) Catholic bishops. The determination as to which of them are or are not in valid orders is up to the Vatican’s Congregation for the Doctrine of the Faith on an individual basis.
Were the Catholic Church switch from the Augustinian Criteria to the Cyprian Criteria for judging the validity of ordinations, this would change, of course, but the Catholic Church has not done that yet.
The Anglo-Lutheran Catholic Church’s orders and sacraments, by the way, are described by Catholic authorities as technically “valid but illicit” though one major Catholic authority has recently been describing its orders and sacraments publicly as “valid but irregular” (which is the same term used for the orders of the PNCC.)
The various Evangelical Catholic Lutheran Churches which have orders and sacraments which are “valid but illicit” / “valid but irregular” in the eyes of the Catholic Church have generally picked up their orders from various Old Catholic Churches, the PNCC via the Nordic Catholic Church, or the Anglo-Lutheran Catholic Church (ALCC) directly or indirectly.
The Church of Sweden has an apostolic succession but as there is no intent to ordain their clergy into a sacerdotal (sacrificing) priesthood, that defect of intent per the Augustinian Criteria would probably result in their orders being ruled invalid by Rome, but unlike the Church of England, the Church of Sweden has not asked the Congregation for the Doctrine of the Faith in Rome for its opinion, so the matter is open. The ALCC re-ordains those clergy who are not ordained in apostolic succession or who come to it with Swedish orders.
This can be a complicated matter, but I hope this helps.
Blessings,
P.S. Please excuse any typos. Proofreading is not my long suit!