What you say is doable but would not be too cost effective.
Having your own box at a co-location centre with a 'fat pipe', or 'pipes', to the internet would be perfectly possible. [Ignoring costs]
You could run your own caching Proxy which you could configure to do whatever stripping of 'cruft' from the traffic you wanted.
Over time the cache would fill and become more effective.
You could also define your own compression methodology, which would be a bit of custom programming for you to do.
(Unless the caching Proxy s/w was particularly sophisticated. Have no recent experience of what is available now.)
You would push as much of the processing as possible to the colo end of the chain.All the above could be done via commercial piece(s) of 'expensive tin'.
Effectively a Private CDN (Content Delivery Network) with compression on the traffic stream. Plus a load of functionality you will probably not use
(Lots of companies provide kit to do all that as specific box or boxes you hang off the back of your WAN and/or in the cloud. Terminology varies but effectively the same thing.)
Question: With all this what would you get in terms of speed-up of browsing etc ?
Question: What is the limit on what you can force down your current connection(s), no matter how much you pre-filter content & compress. ?
Calculate the cost of generating the improved throughput (£/MB/s) vs the cost of more lines multiplexed together (£/MB/s).
Your point of weakness is the colo as in the box which needs some sort of redundancy/failover and the 'Fat Pipes' which needs fallback/failover as well. Plus Administration and Servicing.
Nice Technical Challenge but Budget is a good question and value for money at the end of the day.