From what grounds can an individual Christian read the Bible critically (or not at all)?

It ought to be obvious (as of 2023) that reading the Bible with care and attention - and believing it is 'true' - does not make somebody a Christian; because different people (and different churches) get extremely different things from reading the Bible, or the New Testament, or The Gospels. 

(After all, most of the highest status scriptural scholars have, for several generations, been theological liberals in one or many respects; whose understandings contradict those of the past, and each other; and change through time like any other academic fashion.) 

I have found that (what I regard in myself as) rigorous Christian reflection; has led me to set aside more and more of the Bible as wrong. 

This fact points to the idea of "Christianity without the Bible" - which points at a recognition that the essence of following Jesus may (potentially) be independent of self-identified culture and belief systems adopted in this mortal life. 

As a matter of history, I - and most Christians - start-out with a collection of culturally-injected beliefs about Jesus; many of which are misunderstood, several of which are contradictory, others of which have been deliberately distorted (for evil ends) - and some of which are wholly fabricated. 

Therefore, it is inevitable and necessary that much of developing Christian activity will entail discarding what are recognized as wrong ideas, supposedly derived 'from the Bible'. 

But this kind of attritional process of discarding needs to be based on "Christian principles" - i.e. on some more fundamental and prior understanding of the true nature of Jesus Christ, his work, and his teachings - if it is not to be merely a matter of tailoring Christianity to fit... whatever we happen currently to want in our mortal lives or is expedient from a social perspective (as so obviously happens with "Liberal Christianity").

How, then, can we know of and about Jesus - aside from (and more fundamentally than) church traditions and the Bible? 

There are actually many possibilities for alternative knowledge of Jesus; some of which include: knowledge carried-over from our pre-mortal life as spirits; knowledge from our here-and-now interactions with the Holy Ghost (via prayer and/or meditation); and from 'intuition' - meaning knowledge derived from that-which-is-divine within each Man, as part of our all being Sons and Daughters of God. 

As a back-stop, for those (in the past, as well as present and future) who have zero valid knowledge of Jesus; yet who innately desire his gift of resurrected eternal life in Heaven -- we will all be given a chance to know and follow Jesus after our mortal death.  

So access to the Bible, and an accurate interpretations of its content, is Not essential for salvation. 

Indeed, it would be astonishing if God the Creator (who loves all Men as His children), had made things such that the Bible was essential!

And the same applies to "the church" - whichever church that may be: church is not essential to salvation.

The proper attitude, therefore, is to evaluate and discern whatever is helpful from the Bible and churches - and to reject the rest; from a basis of taking fullest personal responsibility for our faith. And to value and develop our direct and interpersonal ways of attaining valid Christian knowledge - those that are least affected by culture, propaganda, and the errors and evils of other Men.