It’s just that there doesn’t seem to be any actual logic or publicly published citations for why they mark some sources lower or higher.

Their scoring methodology is available here.

then mark them all the way down to Mixed simply due to sources with no examples of said bad sources other than they have a left bias

They actually cite two examples of poorly sourced articles in their analysis summary. One appears to be an opinion piece that they're calling news analysis. The other uses social media as its only sourcing. Reading their MBFC page, there's nothing disqualifying. When reading an article, I'd just pay closer attention to who their sources are.

People (esp. critics) often treat MBFC like it's a binary good/bad indicator. It's really good at that at the edges. It's great at telling you if a source is very good (ex. BBC) or very bad (ex. Sputnik/Fox/InfoWars). The middle is murky. 'Mixed'/'Medium' tells you almost nothing without looking at the reason. It's a wealth of important information for evaluating a source but 'Medium Credibility' should be treated as 'things to keep in mind' or contexts where a source may be less trustworthy. Like, I don't have a problem with this source but I wouldn't trust them to report straight on something that was counter to Luxury Communism as a viable system. I don't think that's an unreasonable position and, again, I don't think I'm worse off for knowing that part of their mission is to promote that ideology. I don't particularly mind CBC News having a higher rating for having no agenda beyond reporting the news.

I don't think MBFC is beyond question but most criticism isn't assessing their strengths and weaknesses in context. It's mostly people taking issue with a single metric (most frequently left/right bias in my experience) and claiming they're "A JOKE" or "HAVE NO CREDIBILITY." And, honestly, 9 times out of 10 it's people who just want to share authoritarian propaganda as news. As I said before, metrics aside, there's a fuckton of valuable information on those pages and no indication that it's not factual. I've never seen an instance where an organization was owned by someone else, based in a different country, or had a different funding model than described. I've never seen someone take issue with their fact-checking. We're definitely not better off not knowing that and there isn't a better alternative.

There’s also others, such as daily mail and Fox News with same factual rating

They specifically list Fox News as a questionable source. They explain the Mixed Factual rating as being because their beat reporting is factual, which is true -- local beat reporters out of, say, Albany aren't far-right goons pushing conspiracy theories. The Daily Mail is rated low credibility and low for factual reporting. You're treating all "Mixed Factual" sources as essentially the same but they're not and aren't intended to be. There are numerous ways to end up with with a Mixed Factual rating, some more serious than others, including uncorrected failed fact-checks, poor sourcing, and lack of transparency by, for example, obfuscating sources by not citing or linking to them. You need to know the reasons and then assess whether it's a dealbreaker.

Zandt has no issues with Times of Israel being left and factual.

They rate the Times of Israel as being center-left. That rating is the least important thing on that site and they're basically trying to do something that's impossible. Left and Right are relative. They feel objective to everyone but vary considerably over time and space. Many (maybe most) Democrats would be Conservatives here in Canada. Being right-wing in the US means something pretty wildly different today than it did 20 years ago. Being "center-left" usually even means something different at the municipal level than federal. There is no universal definition of center-left. That metric is probably most useful at the extremes, but I typically pay little attention to it and wouldn't treat it as anything more than a rough ballpark figure.

Do you have examples of Times of Israel not being factual? People seem to hate them right now because they have "Israel" in their name and it's more asserted as "everyone knows" rather than pointing to any analysis I've seen. I'm not aware of any serious retractions or scandals and I'm almost always able to verify their reporting through other sources (there aren't many English sources reporting on daily goings-on at the Knesset). I did a quick search and couldn't find anything that doesn't reinforce MBFC's assessment. I'd be happy to read whatever you send my way though. Their description of TOI's slight shift toward government perspective since Oct 7 seems accurate. It's also not surprising and happens in every country at war. Ukrainian media, for example, is less critical of the government now than before the full-scale invasion; US media was less critical of the government post-9/11. Most of their reporting is just straight-ahead reporting without editorializing. Looking at their site now, I can't see anything that would even be controversial and their top stories have all been reported by numerous news orgs.