Yes there are studies that investigate this. Before going to literature just some important takeaways:
it’s true that just because employers are de jure required to pay some of the payroll taxes that does not mean that de facto employers actually pay the tax.
Many non-economists hold common misconception that government can assign the incidence of tax burden or that government by declaration can decide what is taxed by a tax it declares. This is in economics known as flypaper theory of tax incidence (see Mankiw. Principles of Economics 8 ed. pp 239), and this theory was never actually even hold by any economist, it's a sort of inside joke to even call it a 'theory'.
however, it’s equally true that usually not all of the burden falls on employees. Only in case the demand for labor is perfectly elastic and labor supply isn’t all of the tax burden would fall on employees. This is highly unrealistic in short-run although in long-run labor demand is probably very elastic.
Both theoretically (ibid. Principles of Economics, ibid Economics of Public Sector, Bradford (1978), Mirrlees & Adam (2010), Chamley (1986) and Judd (1985) - last two papers are technically about capital taxation but they show that any taxes levied on owners are borne by labor in long-run), and empirically (e.g. Roy-Cesar & Vaillancourt (2010), Gruber, 1997 etc.) most of the tax incidence of labor taxes happens to fall on labor supply (i.e. employment) or returns to labor (i.e. wages).
To put some concrete numbers on it a meta-analysis of empirical literature literature by Melguizo and González-Páramo (2012) found that [emphasis mine]:
Based on 52 empirical papers, we find that economic institutions, the tax wedge definition, and the temporal focus significantly affect the results. In the long run, workers bear between two thirds of the tax burden in Continental and Anglo-Saxon economies, and nearly 90 % in the Nordic economies.
Generally, I would say you would be hard pressed to find good estimates that would say that the shifting of the wage burden on employee is less than 50% and generally you will see something around 60-70-80-90% depending on what country we are talking about.
Hence, it would be accurate to say that most but definitely not all of the burden from wage/payroll taxes is shifted to the worker.
This is because demand for workers gets very elastic in the long-run and the distribution of tax burden is to great extent determined by whether supply or demand is more elastic.
Also an important caveat is that this could change in the future. One could argue that one reason why elasticity of demand for labor is so high is that thanks to globalization firms have a lot of options to move production elsewhere. It’s hard to say if world stays globalized after the Covid-19 and Russo-Ukrainian war that disrupted important global supply chains. So it would be worth while to revisit this issue in future, but as of now it’s reasonable to say that most but not all of the tax burden is borne by labor.