Company Logo looks terrible on Modern Site Pages

This is your best bet for branding SharePoint Online sites in O365 from my consulting experience and various clients. And this is coming from a former brand manager myself and a guy who's had to have the predictable debate with marketing/brand teams at various organizations about the 'limitations' of what SPO offers. Most needs of brand teams are wants, not needs. Unfortunately, the system provides restrictive needs, so sometimes you have to stay within those parameters. Hard to say "tough cookies", but sometimes it's the only thing to say.

Here goes:

1. Always--always--add your company's logo to the suite bar and link it to whatever the most popular destination in your network is (usually your intranet home page). Since most logos are more horizontal than vertical, this should cover most companies' brand needs without sacrifice. Do it in PNG or GIF, transparent background, I generally like a white version of any logo on a darker background, but that's just me. This is also a major improvement for your overall navigation experience. Get lost in almost any O365 app? Simply go home!
2. Change the suite bar color to your company's primary brand color.
3. Have a 1:1 default branded icon for SPO sites.
4. For the bigger sites in the network, provide 1:1 graphics that are specific to them, but follow a design motif. What I mean here is have a 1:1 'logo' for the HR, IT, Finance, Facilities, Accounting, Marketing, Production, Legal, etc. sites (and their related sub-sites if you have those). The reason for this recommendation is because you're not going to be able to quickly identify where you are from a generic icon that shows up literally everywhere in SPO. Plus, you know you're in your company's intranet; you don't need a constant reminder in a place that's actually customizable to be helpful in way-finding. Let #1 above do the brand reminding for you. (Note: You can go all out if you want and have one for every intranet and team site, but I'd steer clear of that maintenance mess if you're a big org.)

Looking at your situation specifically:

1. Do #1 above. No question.
2. Consider #2 above if your brand's color palette has an option that isn't obnoxious. Pretty sure it supports hex, so you can get exactly the hue your brand team requires.
3. The flower in your logo is perfect for #3 above. Talk your brand people into it. If they can't be proud of using their logo's icon to meet a 1:1 need (what do they use for Twitter, Facebook, and LinkedIn, by the way?), then they shouldn't be proud of their logo. (Tough love, sorry! This seems like an obvious win to me.)
4. Use the flower icon as the basis of these org icons mentioned in #4 above. Have the flower in the background (50% or more opacity) with shortened names/acronyms in front (100% opacity) to identify which site you're in at a glance.

Some gotchas to keep in mind:

• If you do #1 or #2 above, you need to disable the ability to change the theme of individual sites [Ctrl+F 'Prevent users from overriding custom theming']. Unfortunately, when someone changes the theme (even just the color), it overwrites all of the effects of #1 and #2 in that site. Very annoying, but something you've gotta deal with.
• I don't believe there's a way to set a default image, so you'll have to add the image with every created site (either through some sort of workflow, or include that part in the training for site owners). The default for new sites is the first letter of the first two words of the name, I believe. Which, similar to my comment above about easier identification of a site's name/purpose, was Microsoft's goal here.
• There's no good way I know of to control the images site owners upload. So unless it's a headshot of Darth Vader, I'd say don't worry too much about it, especially on a team site, which should, in theory, have a smaller population to offend with a joke image anyway.

Hope this helps.