It is true to a degree - if your traffic crosses the peering networks between internet service providers, then traffic performance is subject to the vagaries of public traffic loads. However, if all your sites use the same service provider, then your traffic never leaves their network. If, like TalkTalk Business, they are a business-only service provider with strong capacity management policies, then internet VPN performance can rival Managed IPVPN networks.
There are no guarantees, but given the cost benefits of internet access lines compared to other WAN technologies, it is often possible to afford bigger circuits than you might have otherwise considered. This is particularly true of using internet infrastructure to carry traffic internationally, rather than long distance private circuits or Managed IPVPN networks.