Yes it will. If that user uses your service every day, over time, it will be more valuable to them. Most services also don't just stay the same over time. More features are added and bugs are fixed, which adds to the value.
"If they wouldn't pay for it on day 1, the odds are they won't pay for it after 2 years of using the service."
If you never gave them the option of paying in the first place, how do you know?
"I think companies trying this route in the first place will be easier to change than peoples' reaction to it."
The answer is to not have free services in the first place. Many companies have tried the freemium model and then realize that it can't scale (IE: the costs to support this free user base out weighs the benefits).
