I think you are mixing up "neutral" with "apolitical" here - most humanitarian work strives to achieve the former and inherently cannot be the latter!
Take the refugee crisis in the Med as an example: virtually all humanitarian orgs that worked there at the hight of it were hightly critical of the EU, its migration policies, and of Frontex. Because they had to be! Their mission - to save as many lives as possible and safeguard the human rights of the ones that did make it - was simply being hampered by The EU's actions [all of this is still ongoing and still true, by the way].
And, yes, that is political. That does not mean "not neutral" in the sense that it does not mean these orgs are "against the EU" - merely that they, by necessity, oppose the actions and the parts of EU policy that interfere with their own goals. This was not a problem with Ukrainian refugees and: lo and behold, no complaints about the EU then!
Gaza is very comparable in this way!
I suspect that by charities not being "neutral" you are referring to the fact that pretty much every humanitarian organisation on the ground has been heavily critical of Israel's conduct of the war - and, to some extent, of there being a war at all (if you interpret calls for a ceasefire that way).
But, again, they would have to be - because, again, this particular conduct fundamentally hampers (in this particular case: mostly negates altogether) their ability to fulfil their own missions.
Again, this is not biased as such - a "neutrality" problem would be, if the citizens of Tel Aviv were being starved to death and carpet bombed - and MSF refused to provide them with its services. But that is not what is happening. Or if the ICRC (a rather good example in that its mandate explicitly demands neutrality) called for the release of Palestinian prisoners but not Israeli captives (it is actually doing the exact opposite and thereby explicitly not interfering with an Israeli judicial system for Palestinians which is widely criticised - and rightly so! - but that is not what the ICRC does).
I suspect when you call for humanitarian orgs to be "less political" and "more neutral" what you would like to see is less criticism of how the war is conducted and fewer pleas for a ceasefire and accusations that Israel is interfering with humanitarian aid. Again, they cannot do that - they are not stating "Israel bad" but "these conditions make it impossible for us to do our work, and civilians come to unneccessary harm as a result".
Technically, I suppose, they could pivot to what some governments have been doing: drop some food from airplanes or try to establish a sea route. The issue with that is: this is ineffective - and every expert worth their salt says as much! As such - and now call me a cynic - there is also a plausible case to be made that such efforts exist more to a) pander to upset domestic electorates and ... wait for it ... b) political posturing opposite the Israeli government. Neither of which is something that large humanitarian orgs need to worry about (because they are not states or their governments).
None of this is to say there are no flaws in the system - there clearly are. No, I have never been employed by an NGO - I do have some cumulative 3+ years of volunteering in the field under my belt, though, and have seen plenty of stuff I did not think was great. A lot of it, by the way, related to the need to appeal to donors. Organisations demanding change that enables them to do their work, however, is not a "bug" but a "feature" - it doesn't discredit them. If anything it's a testimony to their committment to their actual mission (as opposed to "making no material difference but helping upset Europeans feel a little better").
Oh, also: your Comic Relief bloke you cite? Hardly an expert in the field: he is a business executive with a long history with WPP and precisely none of humanitarian field work who, as one of several such mandates, served on the board of a charity whose main purpose is to funnel funds to other orgs who actually deliver projects - and a card-carrying zionist, so: hardly a beacon of political neutrality himself. HTH.