One major reason that I regard Protestant and Catholic as complementary rather than in competition relates to their specialisms: conversion versus theosis.
Protestants are, or ought to be, experts in evangelism and mission - in getting people 'saved', and 'across the line' into Christianity.
Well, there is a period of strengthening the conversion, learning scripture and removing errors: and this may take many years.
So there tends to be a lot of repetition, and a focus on the moment of conversion, being born again.
The aim is perhaps to secure salvation, to make the state of being saved stronger.
At some point some people begin to want more, or perhaps more accurately want something different (for whatever reason) - and Protestants tend to channel this desire, or this surplus energy, or the need to fill the days with activity into... well, Good Works principally - primarily more evangelism and mission (for which there is always a need) but also into Good Works like health care, social care, education...
The devout Protestant model of church once or twice a week, absent or infrequent (and non-mystical) Holy Communion, perhaps home Bible study groups, and extempore prayer is not well-suited to theosis.
(Protestants tend to be salvation egalitarians, to the point of being reluctant to acknowledge the existence of and gifts of those with exceptional Holiness - for example Saints.)
But for Catholics, conversion is (or ought to be) the start of a process of sanctification (theosis, moving toward being a Saint in communion with God).
And this is aimed at by things such as frequent Holy Communion, ascetic practices, and frequent formal and ritual activity - ideally that kind of fusion of all aspects of life in pursuit of sanctification which reached its highest in Byzantium.
Theosis is, in a way, orthogonal to conversion.
Theosis does not (until its highest levels) render salvation stronger - indeed it tends to risk salvation, by tending to encourage spiritual pride (which can overturn almost any state of salvation - so that very advanced ascetics may fall, presumably into damnation, even after many years of endeavour).
Nonetheless, theosis is probably the main thing which Christians should do after conversion is (reasonably) solid.
On this line of argument, Protestants secure salvation and then some Protestants will want to embark on theosis with a Catholic denomination.
But one major problem is that - while there are numerous successful Protestant evangelicals - there is a serious shortage of Catholic denominations orientated towards theosis. Few Roman Catholic churches seem to offer daily mass. Orthodoxy is weak in the West and does not pervade life. The monastic life is weak and rare. There are few living (or zero) models of advanced sanctification.
But in principle, as an ideal, it would be best (at least in a moderately Christian society) to have both strong Protestant and strong Catholic denominations.