Before narrowing down to asset classes and products within an asset class (i.e. being swayed by marketing), try to be more specific about your liquidity requirements and your downside risk tolerance.
- What percentage of your downpayment savings can you afford to lose before it knocks you out of contention for the price level of house that you hope to buy?
- Assuming the the time between finding a house you want to buy and closing is at most three months, do you really need less than three months' duration for your down payment savings over the next five years? Can you lock the funds up for, say, a year before moving to shorter duration?
Knowing your risk and liquidity profile means that you can more easily choose asset classes, and products therein, based on how they fit your profile. A crude example is below. Your advisor (robo- or not) should have something more sophisticated available.

The way you have posed your questions suggests that you are looking to invest the amount intended for downpayment in a single asset (class) rather than diversifying. In that case you will care about downside risk the most. That could mean ignoring the marketing pitches about potential investment return and sticking to the safest investments that meet your liquidity requirements. Following this reasoning:
Given several years to save for a down payment, are bonds or bond ETFs a smart way to save for a little more % APY? (As opposed to a simple high-yield savings account).
ETFs give you liquidity you don't need right away at the cost of management fees and relatively less certain returns compared with either mutual funds or outright purchases of safe bonds. Yields on funds and ETFs are not directly comparable to savings products where you have limited risk of capital loss, so don't be swayed into thinking that higher yield is always better.
What other performance indicators should be used when evaluating bond ETFs?
For any investment, look at return volatility. It is always backward-looking but that is typically what investment managers target for retail products (as opposed to attempting to achieve X% return and damn the volatility).
Assuming there is another recession in the next few years, how would it affect bonds and bond ETFs? And would that be different for government vs. corporate bonds?
Read the risk factors from some prospectuses. Corporate bond defaults tend to be positively correlated with recessions. Nominal bond yields tend to be positively correlated with inflation; that relationship is sensitive to the proportion of the yield compensating the holder for credit risk.
Would it be better to simply buy the bonds myself?
There are distinct advantages to letting a professional do it for you. Bond index funds exist for that reason.
What advantage does using a bond ETF give?
Liquidity. While they are liquid... but I digress. Read What are the important differences between mutual funds and Exchange Traded Funds (ETFs)?