Electrician recommends to replace 50 amp with 60 amp breaker
Loose connections can cause a lot of problems, including:


Higher power usage - which turns into heat - which can cause a fire (though in this case, the breaker did its job and stopped that)
Arcing - which can cause a fire


So if tightening the connections solved the problem then you should be all set.

In addition, as you suspected, simply replacing a breaker with a larger breaker is not to be done on a whim. It should only be done when:


The wires are able to support the additional current (which is likely NOT the case)
The equipment will be protected properly even with the larger breaker


With a 20-year track record plus tightening connections solving the problem, all indications are that upsizing the breaker would be a very bad thing to do.

The appliance guy blaming the breaker (or more likely "appliance is OK so this must be something in the electrical system") is actually reasonable. An electrician looking at it and recommending replacing a breaker with a larger one based on an overcurrent trip after 20 years is being irresponsible.

Anecdotally, I suspect the appliance guy and the electrician are both getting relatively underpaid by the warranty company - they (generically, not one particular company) are known (at least in my area) to be a bad deal all around - and therefore likely to end up with the "low bidders". Who may not be quite as qualified as you would hope. Just saying...