The avastSvc.exe avast service function may have changed, but not massively - it still controls the various shields and the web shield monitors all http and https (new scan https feature, depending on OS) traffic from your browser. Ordinarily this traffic would be assigned to a browser, but because that traffic is routed through avast so it can be scanned, it is recorded against avastSvc.exe.
There is other traffic related to the File System Shield (also controlled by the avast service), where a check can be made on the avast cloud to see if this file (hash) has any history/reputation to indicate if it may be malicious. This shouldn't put that much on even dial-up. Presumably if you have the dial-up modem off, that check can't be done.
All of this is down to trying to give better protection to the avast user, but it does come with some overheads. Most of the modern top rated AVs also use cloud computing to reference as that is the most up to date information on infections/malware, it isn't just avast doing this.
Avast also does streaming updates so that you get regular signature updates streamed to your system (which can't happen if your modem is off) along with the regular VPS updates - the VPS updates however, are normally done by a different different function instup.exe.
Being stuck on dial-up sucks.