You do not need to reinvent the entire letter for each firm. Your core story, experiences, and reasons for consulting will naturally stay similar because they are about you.
The parts that should change are the ones that claim to explain why that specific firm or office fits. If the McKinsey, BCG, and Bain paragraphs could be swapped without anyone noticing, the tailoring is too shallow.
Good tailoring can be concise. One or two genuinely firm-specific reasons are usually more persuasive than a paragraph filled with slogans from the careers website.
So build a strong base letter, then customize the motivation and any examples that are especially relevant to the firm. That gives you specificity without turning each application into a full rewrite. A useful reference is a McKinsey-specific cover letter guide.