Upvote:3
All the places you mentioned are in the Schengen area and the 90/180 rule applies to the area as a whole. So leaving for three months in the middle is not a small thing you could circumvent by being smart, it's a major condition of visa-free visits designed precisely to preclude what you want to do.
Staying and working for a full 9 months in the Schengen area without a proper long-stay visa is not only illegal, it's also much more difficult to get away with it like many people reportedly do in South America or South-East Asia.
Furthermore, even if you would go out of the Schengen area (say to the UK, Croatia, or even further to Turkey, Georgia, etc.) to avoid falling foul of the rule, you would still not be allowed to work, even remotely. If your clients and your bank accounts are all in the US, you could probably get away with that aspect of your plan but it's still unambiguously illegal.