Is Hamas not the slightest bit culpable for using a hospital to stage militant operations? I think they are.
Is it not a brazen war crime to use the sick and injured as a human shield? In fact, it is.

Hamas militants should be nowhere near any hospital, school, place of worship, or other place where civilians and noncombatants may be gathering. To use places like this as a base of operations is both cowardly and completely against international norms and laws. The IDF would have zero justification for any military operation at Al Shifa Hospital had Hamas not made the horrible decision to use it, and the suffering people within it, as a shield. For context:

The Geneva Conventions of 1949 and their Optional Protocols of 1977 (the main treaties of International Humanitarian Law – IHL) set a range of minimum standards for the conduct of hostilities. They are based upon the fundamental principle of distinction between civilians and other protected persons, on the one hand, and those who take part in hostilities (combatants for short).

The term “civilian” refers to individuals or objects (e.g., premises) that do not have a direct role in hostilities (See Rule 5 and Rule 9 of the Study on customary international law by the International Committee of the Red Cross – ICRC). An attack against a civilian person or object is therefore generally a violation of IHL and may constitute a war crime. A person or object can however lose its civilian status if it starts making an effective contribution to military action. It would then become a legitimate military objective (and hence a target) (See Rule 10 of the Study on customary international law by the ICRC). This determination must however be unequivocal: when in doubt as to whether a school or hospital has become a military objective, there is a presumption that it retains its civilian status.

Even attacks against legitimate military targets must, however, follow two additional principles: 1) the principle of proportionality – whereby an attack that would cause incidental loss of civilian life, injury to civilians, damage to civilian objects, or a combination thereof, which would be excessive in relation to the concrete and direct military advantage anticipated, is prohibited (See Rule 14 of the Study on customary international law by the ICRC) – and 2) the principle of precaution in attack – which states that constant care must be taken to spare the civilian population, civilians and civilian objects. All feasible precautions must be taken to avoid, and in any event to minimize, incidental loss of civilian life, injury to civilians and damage to civilian objects (See Rule 15 of the Study on customary international law by the ICRC).

https://watchlist.org/publications/what-does-international-law-say-about-attacks-on-schools-and-hospitals/

Hamas are certainly not ignorant of this, and so they must know that by using the Al Shifa Hospital as a strategic location they are deliberately putting a target on it, so why would they do it?

When asked by reporters at the White House what his message to family members of hostages was, he said: "Hang in there, we're coming." [...]

White House spokesman John Kirby told reporters on the presidential plane, Air Force One, that intelligence confirmed the militant Hamas group, which rules Gaza, used tunnels underneath Al-Shifa and other hospitals to conceal military operations and to hold hostages.

• U.S. backs claim Hamas uses Gaza hospitals as military cover amid hopes for hostages' release

I'm not sure how you interpret "we're coming" by Biden regarding hostages that US intelligence says are being held in the hospital currently being raided, but it certainly doesn't seem like Israel is doing this at odds with Biden's intentions.

If anything, Biden's red line of "hospitals must be protected" seems like it might be behind the IDF claim in the original article of:

The IDF forces include medical teams and Arabic speakers, who have undergone specified training to prepare for this complex and sensitive environment, with the intent that no harm is caused to the civilians.

We'll see what the next few hours hold, but Biden's comments just a few hours before and the US intelligence publicly weighing in certainly looks like this is intended to be a hostage rescue operation which the US is at very least aware of if not actively involved in.