I realize how tempting it is to lump it all under 'writing software' and calling it a day, but the culture at a services company is very different from a product company, in my experience. Plus it's really hard to give your customers a USP when you're essentially saying 'we do everything!' which you don't, unless you're 5k people big.
"From a company and team perspective I see no problem in doing both: services and products."
Yeah I'm going to have to disagree on this one. Sure it's fine in the beginning, but it's just a major pain once you start having regular release cycles and project deadlines that don't match, conflicts when some employees are more 'service providers' and others more 'product builders' (the two require different personalities, imo), cash flow and compensation differences and many more.
There are many companies who were in this position once, and all that I know made a choice of one or the other (usually product, that's where the money is if the product is good; services is a cost plus business). There were a few talks on this topic on the BoS conference in Dublin earlier this year, it's probably a recurring theme - I'm sure there have been more talks on it over the years.