This is tangential, but part of the reason why I dislike talking to servers about the tipped minimum wage. Because it seems they deliberately don't want to understand the full context of the law. Or fully the type of work they are in.
What you are describing is typically called "side work". Basically work that is not directly related to waiting a table. Folding napkins, putting up chairs, moving tables, etc. Even in cases where the restaurant clocks in their workers for every minute they are doing work, servers complain about doing side work because "it costs them money". And they want the federal minimum wage during that time.
Which naively makes sense. But that's not at all how tipped wages work under the law. The formula is basically: (([hours worked] * [tipped minimum wage]) + [tips earned in pay period]) / [hours worked] = [hourly wage over pay period]
So if you work 10 hours in a pay period, that's $21.30. If you collected $51.20 in tips during that time, you made $72.50 or $7.25/hr (based on current federal minimum wage, obviously numbers are different for different areas where the minimum wage is different from the federal). The restaurant is in the clear. Even if you worked 5 hours doing "side work" and 5 hours waiting tables where you got $51.20 in tips.
Servers want to essentially prorate every minute they are not physically collecting a tip it seems. Worked an hour and every table stiffed you? They want $7.25 for that hour. Regardless of the fact that the previous hour they took in $20 in tips.
Serving is more akin to being an independent contractor. You need to make hay while the sun is shining. The fat sustains you through the lean. You look at your per hour over the aggregate, not counting every hour. It's why contractors charge the rates they charge. If you are only billing half the year, those billings need to support you through the other half.
Now, in your case, the reason they don't want you to clock in isn't to save $7 over a week. It's to up your average. Which is still wrong. Let's say at your current rate of collecting tips, you totaled $225 over 30 hours. That's $7.50. Restaurant doesn't owe you anything. Now, $225 over 32.5 hours (your low end over 5 days), that's only $6.92 / hour. The restaurant is short, not $7, but $41.28
And true, If you instead made $300 over those 30 hours, the extra 2.5 hours don't even matter to you. Your check wouldn't have been that different. But, it's money that's not coming from their pockets and it's really to push people near the bubble over their limit.