A practical test: write three lists—costs due immediately, work that can wait a year, and permanent compromises. Put moving, the first mortgage payment and any urgent inspection items in the first; decoration and most furniture in the second; commute and ongoing service charges in the third. If...
I understand why the emergency-fund test is reassuring, but I would hesitate to let it answer the whole question. Cash left after the first mortgage payment and immediate repairs can absorb a surprise; it cannot shorten the commute or reduce an ongoing commitment that already feels...
Can the agent explain why shortening the inspection window would strengthen this particular offer? If there is no stated seller concern about timing, you may be conceding something without gaining much.