Don't look at the stats.
You'll obviously remember that Biagini was fantastic as a reliever in 2016. He had a bag of pitches too. People thought he could probably start.
You'll also clearly remember that he ended up being awful in 2017 as a starter.
Look at the depth chart.
The first four SP are clear: Stroman, Sanchez, Estrada, Happ.
The bullpen is not clear: Osuna, Barnes, Dermody, Tepera, Loup, Leone, Carlos Ramirez. Gross - Sure there are a few intriguing arms but after Osuna every name there could conceivably be replacement level.
Immediate minor league options for the rotation include: Borucki, Guerrieri, Pannone. Decent but not well developed.
Immediate options for the bullpen: Greene, Reid-Foley (possible SP later), Rios, Rowley, Mayza, Girodo.
Shittier rotation options: Harris, McGuire.
It doesn't seem like Zeuch or Romano are realistic options in 2018.
Clearly a #5 SP would be sweet. Clearly a good bullpen arm would be sweet. There is need on both sides. Toronto's pen is projected 24/30 and their rotation is 16/30.
Look at the stats.
Biagini started 18 games in 2017 and relieved in 26.
He contributed 1.4 fWAR in 2017 in 119.2 innings, compared to 1.2 in 67.2 relief innings in 2016.
He had a 5.34 ERA in 2017; 5.73 as a SP and 4.26 as a reliever.
He had a 4.27 FIP in 2017; 4.36 as a SP and 4.01 as a reliever. xFIP was 4.23 as SP vs 3.79 as a reliever.
He had a 10.6 K-BB% in 2017; 10.2% as a SP and 11.9% as a reliever.
He had a HR/9 of 1.13 in 2017 compared to just 0.40 in 2016. He had a 1.23 mark as a SP and 0.85 as a reliever.
Discussion and Conclusion
The evidence against Biagini as a starter amounts to a small sample HR/FB and ERA blip. When we look at K and BB rates and DIPS, he was a relatively effective starter.
He also wasn't that much better as a reliever. His 2016 relief performance didn't really pass the sniff-test anyway; nobody should have assumed he was a top-flight reliever based on that one season.
A DIPS mark in the low-mid 4's equates to an average or better SP in the AL East, if the player can reliably accumulate innings. See these examples:
http://www.fangraphs.com/statss.aspx?playerid=2717&position=P
http://www.fangraphs.com/statss.aspx?playerid=12917&position=P
http://www.fangraphs.com/statss.aspx?playerid=14107&position=P
There's not enough information to give up on Biagini as a starter. He still has enough intriguing characteristics to give him the chance: he seems durable, he has four pitches, his velocity transitioned fine as a SP (sitting 92/93).
The team would benefit tremendously from the SP depth so players like Pannone, Borucki, and Guerrieri can get some AAA innings. The rotation would be set and the brass could focus on RP additions. Players like Greene and Reid-Foley could be given legitimate RP tryouts in spring training. Also, any "boost" Biagini would give the bullpen is probably not as significant as some might think/assume.