Well - that was generally the case and still often is, but also things have changed a bit.
McKinsey, BCG and Bain have grown so large that they basically offer "everything" now, not just the strategy part, but in many cases also at least parts of the implementation. Clients have found that it also creates more value this way by keeping a small MBB team on the ground and not losing access to all the knowledge that was generated.
So quite honestly, the key difference between "strategy" consultancies and more implementation-focused firms is the daily rate they charge. (MBB would argue the value is different, too.) If clients are willing to pay MBB's high rates, they can get as many consultants for as long as they want :) Large, company-wide "transformation" projects with >50 MBB consultants are still the exception, but have grown far more common over the past years.
For young consultants and associates, this means they can get stuck in a PMO (project management office) in the middle of a 3-year-transformation case, where the main task is to check project progress and update traffic light slides. Consultants hate it, but actually, in terms of sleep and work-life-balance, those projects can be a bit of an oasis compared to the deserts of 3 months high-stakes strategy cases with a small team and competive consulting firms working next door, or a 2 week due diligence case with neither sleep nor weekend :)
Regarding other strategy firms, you have (among others) Oliver Wyman, AT Kearney, Roland Berger (in Europe) and the strategy shops of the big 4 accounting firms (strategy& - formerly Booz, DeloitteMonitor, etc.). Some lists here: https://www.quora.com/What-are-the-top-tier-management-consulting-firms
In addtition, there are tons of smaller top notch firms specialized in certain industries, regions, or topics.