1. Using the web would become much more compute/energy intensive and old devices would quickly lose access to the modern web.
2. Some hosts would inevitably double-dip by implementing this and ads or by "overcharging" the amount of work. There would have to be some kind of limit on how much work can be required by hosts - or at least some way to monitor and hold hosts accountable for the amount of work they charge.
3. There would need to be a cheap and reliable way to prove the client's work was correct and accurate. Otherwise people will inevitably find a way to spoof the work in order to reduce their compute/energy cost.
All you need is a central clearing house service that can handle billions of 0.000001 transactions per day.
Incidentaly, I doubt the bitcoin chain could handle that...
It's a solution that already has adoption, does not require everyone to sign up with a centralized service, and does not require everyone to pay money (they can pay with small amounts of computation instead) so it remains accessible to ~everyone.
Of course it isn't very secure because if the client sees a mined block they might have the technical savvy keep it. But you'd be forcing big web scrapers to run a horribly inefficient mining operation and they'd hate it. Plus you can run a blacklist of hated clients and double the difficulty for them, which is very low-cost for false positives and very high-cost for real scrapers - that isn't a result of using Bitcoin but it'd be funny.