The first approach offers reasonable odds of success, but can suffer because even though a supplier has a good reputation, it does not mean they will be good for your business. An alternative is to employ a consultant to select a supplier. However, apart from the expense, does the consultant have an unbiased view or are they affiliated to a specific supplier?
The next step is to decide who is going to be involved in making the decision. One possibility is to limit the involvement to key managers and users. This will enable them to apply their knowledge and experience to assess the functional fit of the software to the needs of your organisation.
The person dealing with technical issues may want to be involved in the early stage of supplier evaluation, as they will be maintaining the system after it has been installed.
A small team of managers and users could sit in on product demonstrations and then meet to agree a shortlist. A key constraint is time. The team of managers and users should be in the selection process when there are only three or four suppliers.