How to remove iCloud tabs from iPhone suggested in Safari on Mac?

This may not be the solution you are seeking, but I will toss it out there anyway.

In the lower right corner of the screen, there is a 3 line / 3 dot icon. Click on it and one of the options available is to turn off the "iCloud Tabs" such that no longer appear on the page at all. This doesn't require that you stop syncing. Just stops showing them.

No worries. Those tick boxes control what you want to appear on the Start Page screen. The control of sync is in you iCloud settings.

Regarding your "old irrelevant links" I did some testing with my iPhone and iPad to see what appears. I can confirm that what I am seeing is likewise not up to date. For both devices, the tabs showing up are of closed websites. More problematic is that a currently open tab on the devices does not show up at all. Presumably, there is some level of delay in these updating.

You said "flooding my homepage with old useless suggestions" which is interesting. For both my iPhone, and iPad, I see only one suggested tab. Purely a guess on my part, but since we are concluding that the suggested tabs are NOT representative of what is currently open on the devices, there must be some snapshot in time. Presumably, your snapshot was at a time when you had many tabs open all at the same time? Does that seem to be the case?

Again, purely conjecture on my part, but it would seem that this tab snapshot is likely out on iCloud and not a function of browser history in any of the actual devices. Thus why clearing browser history is not clearing out the suggested tabs.

Yep. Looking at this: https://support.apple.com/en-us/HT202530 The wording is very specific in saying "open tabs from your other supported devices." Obviously, not currently the case. Likely (hopefully) just a transient bug.