There is nothing wrong with running a practice on spreadsheets. Plenty do it well. The question is not whether spreadsheets work, but when they stop being the cheaper option.
Sign one — the list has a keeper
When one person maintains the deadline sheet, the practice has a dependency rather than a system. Holidays become risk.
Sign two — you reconcile between sheets
The deadline sheet, the job sheet and the billing sheet stop agreeing. Time gets spent making them agree, and that time is pure loss.
Sign three — you cannot answer capacity questions
"Can we take this client on?" becomes a judgement call rather than a look at workload. That is when growth starts feeling risky.
Sign four — billing lags the work
Work finishes, and the invoice happens weeks later because assembling what to bill requires a conversation.
Sign five — nobody knows what jobs cost
Fixed fees get repeated year after year without anybody testing them against the hours. This is usually the most expensive of the five and the least visible.
What changes
Software does not make the work quicker. It makes the coordination around the work cheaper — the tracking, the allocating, the assembling, the checking. In most practices that coordination is the larger cost.