Opinions are valuable, but when they are based on errors, they become dangerous.  Following are the disparities, the claims of pundits and my truths:

Claim: 4 core vendors owned 96% of the market in 2013.
Truth: The four never owned that much. In 2013, "Automation in Banking" reported 81% was owned by Fiserv, JackHenry, FIS and Harland Financial Solutions. They were the Big 4 in 2013.
Claim: Now, the 4 own 76%.
Truth: Top 4 currently own 78% - Fiserv, JackHenry, FIS, FinastraUSA.
Claim:  New core vendors (startups) caused the decline from 81% to 76%.
Truth:  That's a dumb claim. New core vendors don't even own 1% of the market today.
Claim:  The perceived success of startups is based on announcements of huge investments.
Truth:  My stats are based on new core sales to US banks and credit unions, not investments in futures.
Claim:  The top 4 US vendors are Fiserv, JackHenry, FIS & FinastraUSA
Truth:  Correct, as long as one realizes that Finastra is a Global vendor (aka International financial institutions). Finastra's US customer base is still HFS with no core growth in the US since it was acquired by Finastra.
Claim: Top 4 started to innovate as a defensive move against startups.
Truth:  The predecessors of the top 4 (M&I Data Services, Computer Services, Inc, Florida Software, NCR and Fiserv) invented the idea of development, maintenance and innovation from the beginning of the bank tech industry - 1964.  They knew the meaning of customer satisfaction: Send out fresh software updates and FIs will automatically evolve with the times, and they will rely on their primary vendor to keep them current.
Claim:  Pundits use "legacy" as if it were a dirty word.
Truth:  Astute bankers are using their core systems because bank employees become more proficient the longer they stick to what they know.  Also, vendors have added functionality to their software that make it richer, and more adaptive to regulatory demands. Fiserv, FIS and JackHenry continue to score highest performance in new sales because of their track record during the past 55 years.
Claim:  Pundits report only about the top 4 banks in the US.
Truth:  There are 9 smaller core vendors that own 21% collectively of the market, and their performance is stronger than any startup. Nine other companies account for the whole (22) but they do not report their numbers.  Reliance on 99% of marketshare is a guarantee that bankers enjoy just like they enjoy borrowers who pay off their loans.

The above analysis relates only to US core systems.  There are hundreds of ancillary applications, some with independent IT companies (best-of- breeds), others developed by the core vendors.  Several successful independents have been acquired by the big cores.  Recently D3, Geezeo, Malauzai, First Data and Worldpay are examples.  It is a strategy that the Big 4 have exercised with success. There are others in the due diligence stage that I don't know about thanks to "quiet period" regulations.  Going from startup to ripe acquisition requires a dozen years of successful growth.

Here are my thoughts regarding core system selection:

Go with a Top 3                                very little risk
Force a Big 4 to go with a Best-of Breed-ancillary        big risk
Go with a startup (anyone with an idea)                              the biggest risk
Wait for a startup to become a reliable resource        too long
Wait until the dust settles                        most common

The facts and opinions reported in this document represent research I conducted starting in 1988.  The information has been refreshed and published annually in a report called, "Automation in Banking."  My customers include, Financial Institutions, Bank-Tech Vendors, Investors, Attorneys, Regulators, and Consultants.  Each year, I have asked them for suggestions that improve the content.  One of my customers from IBM suggested a one-page graphic showing the functionalities of each vendor, 546 pages was too much to grasp at a glance.  I gladly complied, as did the vendors in the report.