No, it wouldn't. It would be exactly the same. Because their employer knows exactly how much they will get in tips and subtracted that same amount from their salary. The cost of labor is constant, regardless of how you divide it.
If servers were paid entirely by the employers directly, they'd be paid the least amount that got an acceptable level of employee. Same as all other service industry jobs.
Since the payment was coming from the employer, it would be included in the meal price. Meal price is a competitive factor for restaurants, and all else being equal, those restaurants with lower meal prices will do better than equivalent restaurants with higher meal prices. The pressure to have lower meal prices would lead to lower pay for the employees. That would be one of the easiest places to cut costs, and thus prices.
Customers don't, as a rule, include tips in their evaluation of meal prices. So it removes the downward competitive pressure there.
What this really boils down to is that customers are more generous with direct payments to servers than their employers would be. Do you think this isn't true? Why?
Plenty of study shows that generosity or level of service have very little to do with tipping on the whole. People typically tip based on personal habit relative to their interpretation of local custom. (I can try to find sources if you doubt this.) I don't think there's any reason to believe that customers don't take gratuity, and tax for that matter, into account when deciding when and where to dine out.
I'm not suggesting that the situation would be the exact same for every server if tipping went away. Some would benefit, others would suffer. On the whole, it would be less discriminatory. But I don't see why we would expect to see a macro change in waitstaff income.